Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A

Percentage Accurate: 99.3% → 99.3%

Time: 6.0s

Alternatives: 16

Speedup: 1.0×

• 44.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{t \cdot t}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (sqrt (exp (* t t)))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * sqrt(exp((t * t)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * sqrt(exp((t * t)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.sqrt(Math.exp((t * t)));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.sqrt(math.exp((t * t)))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * sqrt(exp(Float64(t * t))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * sqrt(exp((t * t)));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Exp[N[(t * t), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   5. exp-sqrt N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   7. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

   8. exp-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   10. lower-exp.f64 99.4

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

4. Applied rewrites 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{{\left(e^{t}\right)}^{t}}} \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   3. pow-exp N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{e^{t \cdot t}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{{t}^{2}}}} \]

   5. lower-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{e^{{t}^{2}}}} \]

   6. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

   7. lift-*.f64 99.3

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

6. Applied rewrites 99.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{e^{t \cdot t}}} \]
7. Add Preprocessing

Alternative 2: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)\\ t_2 := x \cdot 0.5 - y\\ \mathbf{if}\;t \leq 95:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{z \cdot 2}\right) \cdot t\_1\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+46}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(t\_2 \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1
         (fma
          (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
          (* t t)
          1.0))
        (t_2 (- (* x 0.5) y)))
   (if (<= t 95.0)
     (* (* t_2 (sqrt (* z 2.0))) t_1)
     (if (<= t 4e+46)
       (* (* (sqrt (* (* 2.0 z) (pow (+ 1.0 t) t))) 0.5) x)
       (* (* t_2 (sqrt (* (* (sqrt (* 2.0 z)) (sqrt 2.0)) (sqrt z)))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0);
	double t_2 = (x * 0.5) - y;
	double tmp;
	if (t <= 95.0) {
		tmp = (t_2 * sqrt((z * 2.0))) * t_1;
	} else if (t <= 4e+46) {
		tmp = (sqrt(((2.0 * z) * pow((1.0 + t), t))) * 0.5) * x;
	} else {
		tmp = (t_2 * sqrt(((sqrt((2.0 * z)) * sqrt(2.0)) * sqrt(z)))) * t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0)
	t_2 = Float64(Float64(x * 0.5) - y)
	tmp = 0.0
	if (t <= 95.0)
		tmp = Float64(Float64(t_2 * sqrt(Float64(z * 2.0))) * t_1);
	elseif (t <= 4e+46)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * (Float64(1.0 + t) ^ t))) * 0.5) * x);
	else
		tmp = Float64(Float64(t_2 * sqrt(Float64(Float64(sqrt(Float64(2.0 * z)) * sqrt(2.0)) * sqrt(z)))) * t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision]}, If[LessEqual[t, 95.0], N[(N[(t$95$2 * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 4e+46], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[Power[N[(1.0 + t), $MachinePrecision], t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[(t$95$2 * N[Sqrt[N[(N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 95

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

      14. lift-*.f64 96.0

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

   5. Applied rewrites 96.0%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]

   if 95 < t < 4e46

   1. Initial program 98.9%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. lower-+.f64 73.7

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 73.7%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot {\left(1 + t\right)}^{t}} \cdot 0.5\right) \cdot x \]

   if 4e46 < t

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      2. rem-square-sqrt N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\sqrt{z \cdot 2} \cdot \sqrt{z \cdot 2}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      3. sqrt-prod N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\sqrt{z \cdot 2} \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      4. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\sqrt{z \cdot 2} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{z}\right)}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{2}\right) \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{2}\right) \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{\left(\sqrt{z \cdot 2} \cdot \sqrt{2}\right)} \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      8. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\color{blue}{\sqrt{z \cdot 2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{\color{blue}{z \cdot 2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{\color{blue}{z \cdot 2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      11. *-commutative N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{\color{blue}{2 \cdot z}} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{\color{blue}{2 \cdot z}} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      13. lower-sqrt.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \color{blue}{\sqrt{2}}\right) \cdot \sqrt{z}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

      14. lower-sqrt.f64 99.3

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \color{blue}{\sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}}\right) \cdot e^{\frac{t \cdot t}{2}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right) \]

      4. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

      5. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

      7. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      9. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      11. pow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

      13. pow2 N/A

          \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

      14. lift-*.f64 98.8

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

   7. Applied rewrites 98.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\left(\sqrt{2 \cdot z} \cdot \sqrt{2}\right) \cdot \sqrt{z}}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\left(t \cdot t\right) \cdot 0.5} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (* (* t t) 0.5))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) * 0.5));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * exp(((t * t) * 0.5d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.exp(((t * t) * 0.5));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.exp(((t * t) * 0.5))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * exp(Float64(Float64(t * t) * 0.5)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * exp(((t * t) * 0.5));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   5. exp-sqrt N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   6. pow1/2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{{\left(e^{{t}^{2}}\right)}^{\frac{1}{2}}} \]

   7. exp-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{{t}^{2} \cdot \frac{1}{2}}} \]

   8. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{1}{2} \cdot {t}^{2}}} \]

   9. lower-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{1}{2} \cdot {t}^{2}}} \]

   10. *-commutative N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{{t}^{2} \cdot \frac{1}{2}}} \]

   11. lower-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{{t}^{2} \cdot \frac{1}{2}}} \]

   12. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{2}} \]

   13. lift-*.f64 99.3

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\left(t \cdot t\right)} \cdot 0.5} \]

4. Applied rewrites 99.3%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\left(t \cdot t\right) \cdot 0.5}} \]
5. Add Preprocessing

Alternative 4: 86.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{if}\;t \leq 5200:\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot t\_1\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{+132}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t\_1, t \cdot t, 1\right)} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* t t) 0.5 1.0)))
   (if (<= t 5200.0)
     (* (* (- (* x 0.5) y) (sqrt (+ z z))) t_1)
     (if (<= t 1.95e+63)
       (* (sqrt (* 2.0 z)) (- (* (- (/ y x) 0.5) x)))
       (if (<= t 1.55e+132)
         (* (* (sqrt (* (* 2.0 z) (fma t_1 (* t t) 1.0))) 0.5) x)
         (* (sqrt z) (* (* (* t t) 0.5) (* (- (* 0.5 x) y) (sqrt 2.0)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t * t), 0.5, 1.0);
	double tmp;
	if (t <= 5200.0) {
		tmp = (((x * 0.5) - y) * sqrt((z + z))) * t_1;
	} else if (t <= 1.95e+63) {
		tmp = sqrt((2.0 * z)) * -(((y / x) - 0.5) * x);
	} else if (t <= 1.55e+132) {
		tmp = (sqrt(((2.0 * z) * fma(t_1, (t * t), 1.0))) * 0.5) * x;
	} else {
		tmp = sqrt(z) * (((t * t) * 0.5) * (((0.5 * x) - y) * sqrt(2.0)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t * t), 0.5, 1.0)
	tmp = 0.0
	if (t <= 5200.0)
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * t_1);
	elseif (t <= 1.95e+63)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(-Float64(Float64(Float64(y / x) - 0.5) * x)));
	elseif (t <= 1.55e+132)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t_1, Float64(t * t), 1.0))) * 0.5) * x);
	else
		tmp = Float64(sqrt(z) * Float64(Float64(Float64(t * t) * 0.5) * Float64(Float64(Float64(0.5 * x) - y) * sqrt(2.0))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, If[LessEqual[t, 5200.0], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[t, 1.95e+63], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-N[(N[(N[(y / x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 1.55e+132], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], N[(N[Sqrt[z], $MachinePrecision] * N[(N[(N[(t * t), $MachinePrecision] * 0.5), $MachinePrecision] * N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if t < 5200

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      4. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      5. exp-sqrt N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      7. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

      8. exp-prod N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

      10. lower-exp.f64 99.4

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

   4. Applied rewrites 99.4%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{{\left(e^{t}\right)}^{t}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      5. lift-*.f64 90.0

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   7. Applied rewrites 90.0%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      3. count-2-rev N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      4. lower-+.f64 90.0

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   9. Applied rewrites 90.0%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   if 5200 < t < 1.95e63

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 15.2

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      6. lower-/.f64 24.9

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   8. Applied rewrites 24.9%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   if 1.95e63 < t < 1.5499999999999999e132

   1. Initial program 99.5%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + \frac{1}{2} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + \frac{1}{2} \cdot {t}^{2}, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \frac{1}{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({t}^{2}, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 70.5

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 70.5%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   if 1.5499999999999999e132 < t

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

      3. distribute-rgt-out N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   5. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]

   6. Taylor expanded in t around inf

      \[\leadsto \sqrt{z} \cdot \left(\frac{1}{2} \cdot \color{blue}{\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left({t}^{2} \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      4. pow2 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right)\right)\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \left(\color{blue}{\frac{1}{2}} \cdot x - y\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\left(\frac{1}{2} \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

      11. lift-sqrt.f64 97.3

          \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2}\right)\right) \]

   8. Applied rewrites 97.3%

      \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(0.5 \cdot x - y\right) \cdot \sqrt{2}\right)}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ t_2 := \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot t\_1\\ \mathbf{if}\;t \leq 5200:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right)\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+133}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t\_1, t \cdot t, 1\right)} \cdot 0.5\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* t t) 0.5 1.0))
        (t_2 (* (* (- (* x 0.5) y) (sqrt (+ z z))) t_1)))
   (if (<= t 5200.0)
     t_2
     (if (<= t 1.95e+63)
       (* (sqrt (* 2.0 z)) (- (* (- (/ y x) 0.5) x)))
       (if (<= t 3.1e+133)
         (* (* (sqrt (* (* 2.0 z) (fma t_1 (* t t) 1.0))) 0.5) x)
         t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t * t), 0.5, 1.0);
	double t_2 = (((x * 0.5) - y) * sqrt((z + z))) * t_1;
	double tmp;
	if (t <= 5200.0) {
		tmp = t_2;
	} else if (t <= 1.95e+63) {
		tmp = sqrt((2.0 * z)) * -(((y / x) - 0.5) * x);
	} else if (t <= 3.1e+133) {
		tmp = (sqrt(((2.0 * z) * fma(t_1, (t * t), 1.0))) * 0.5) * x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t * t), 0.5, 1.0)
	t_2 = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * t_1)
	tmp = 0.0
	if (t <= 5200.0)
		tmp = t_2;
	elseif (t <= 1.95e+63)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(-Float64(Float64(Float64(y / x) - 0.5) * x)));
	elseif (t <= 3.1e+133)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t_1, Float64(t * t), 1.0))) * 0.5) * x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[t, 5200.0], t$95$2, If[LessEqual[t, 1.95e+63], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-N[(N[(N[(y / x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision])), $MachinePrecision], If[LessEqual[t, 3.1e+133], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t$95$1 * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if t < 5200 or 3.1e133 < t

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

      4. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

      5. exp-sqrt N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

      7. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

      8. exp-prod N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

      9. lower-pow.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

      10. lower-exp.f64 99.3

          \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{{\left(e^{t}\right)}^{t}}} \]

   5. Taylor expanded in t around 0

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

      4. pow2 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      5. lift-*.f64 90.8

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   7. Applied rewrites 90.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      2. *-commutative N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      3. count-2-rev N/A

         \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

      4. lower-+.f64 90.8

         \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   9. Applied rewrites 90.8%

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

   if 5200 < t < 1.95e63

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 15.2

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 15.2%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      6. lower-/.f64 24.9

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   8. Applied rewrites 24.9%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   if 1.95e63 < t < 3.1e133

   1. Initial program 99.6%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 76.0%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + \frac{1}{2} \cdot {t}^{2}\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + \frac{1}{2} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + \frac{1}{2} \cdot {t}^{2}, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \frac{1}{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left({t}^{2}, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 70.2

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 70.2%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 94.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma
   (fma (fma 0.020833333333333332 (* t t) 0.125) (* t t) 0.5)
   (* t t)
   1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(fma(fma(0.020833333333333332, (t * t), 0.125), (t * t), 0.5), (t * t), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(fma(0.020833333333333332, Float64(t * t), 0.125), Float64(t * t), 0.5), Float64(t * t), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.020833333333333332 * N[(t * t), $MachinePrecision] + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right), \color{blue}{{t}^{2}}, 1\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

   5. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right) \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

   7. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48} \cdot {t}^{2} + \frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, {t}^{2}, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   9. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   11. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   13. pow2 N/A

       \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{48}, t \cdot t, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

   14. lift-*.f64 94.7

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

5. Applied rewrites 94.7%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.020833333333333332, t \cdot t, 0.125\right), t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
6. Add Preprocessing

Alternative 7: 92.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma (fma 0.125 (* t t) 0.5) (* t t) 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * fma(fma(0.125, (t * t), 0.5), (t * t), 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(fma(0.125, Float64(t * t), 0.5), Float64(t * t), 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}\right) \cdot {t}^{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{2} + \frac{1}{8} \cdot {t}^{2}, \color{blue}{{t}^{2}}, 1\right) \]

   4. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\frac{1}{8} \cdot {t}^{2} + \frac{1}{2}, {\color{blue}{t}}^{2}, 1\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, {t}^{2}, \frac{1}{2}\right), {\color{blue}{t}}^{2}, 1\right) \]

   6. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right) \]

   8. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{8}, t \cdot t, \frac{1}{2}\right), t \cdot \color{blue}{t}, 1\right) \]

   9. lift-*.f64 92.6

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot \color{blue}{t}, 1\right) \]

5. Applied rewrites 92.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right), t \cdot t, 1\right)} \]
6. Add Preprocessing

Alternative 8: 87.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (fma (* t t) 0.5 1.0) (* (fma 0.5 x (- y)) (sqrt 2.0))) (sqrt z)))

C

double code(double x, double y, double z, double t) {
	return (fma((t * t), 0.5, 1.0) * (fma(0.5, x, -y) * sqrt(2.0))) * sqrt(z);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(fma(Float64(t * t), 0.5, 1.0) * Float64(fma(0.5, x, Float64(-y)) * sqrt(2.0))) * sqrt(z))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision] * N[(N[(0.5 * x + (-y)), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Sqrt[z], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \color{blue}{\sqrt{z}} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)\right) \cdot \sqrt{z} + \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \cdot \color{blue}{\sqrt{z}} \]

   3. distribute-rgt-out N/A

      \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{z} \cdot \color{blue}{\left(\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right) + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{z} \cdot \left(\color{blue}{\frac{1}{2} \cdot \left({t}^{2} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)\right)} + \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{z} \cdot \left(\left(\frac{1}{2} \cdot {t}^{2}\right) \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) + \color{blue}{\sqrt{2}} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\frac{1}{2} \cdot {t}^{2}, \color{blue}{\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)}, \sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right) \]

5. Applied rewrites 87.8%

   \[\leadsto \color{blue}{\sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right)} \]
6. Step-by-step derivation

   1. pow2 87.8

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \]

   2. exp-sqrt-rev 87.8

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \]

   3. pow2 87.8

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \]

   4. pow-exp 87.8

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\left(t \cdot t\right) \cdot 0.5, \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(0.5, x, -y\right)\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \sqrt{z} \cdot \color{blue}{\mathsf{fma}\left(\left(t \cdot t\right) \cdot \frac{1}{2}, \sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right)} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{z} \cdot \mathsf{fma}\left(\color{blue}{\left(t \cdot t\right) \cdot \frac{1}{2}}, \sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right), \sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) \]

   7. lift-fma.f64 N/A

      \[\leadsto \sqrt{z} \cdot \left(\left(\left(t \cdot t\right) \cdot \frac{1}{2}\right) \cdot \left(\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)\right) + \color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(\frac{1}{2}, x, -y\right)}\right) \]

7. Applied rewrites 87.8%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(t \cdot t, 0.5, 1\right) \cdot \left(\mathsf{fma}\left(0.5, x, -y\right) \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}} \]
8. Add Preprocessing

Alternative 9: 66.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{2 \cdot z}\\ \mathbf{if}\;t \leq 6.9 \cdot 10^{-6}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+63}:\\ \;\;\;\;t\_1 \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* 2.0 z))))
   (if (<= t 6.9e-6)
     (* t_1 (fma 0.5 x (- y)))
     (if (<= t 7.5e+63)
       (* t_1 (- (* (- (/ y x) 0.5) x)))
       (* (* (* (* t t) (* (sqrt (* z 2.0)) 0.5)) 0.5) x)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((2.0 * z));
	double tmp;
	if (t <= 6.9e-6) {
		tmp = t_1 * fma(0.5, x, -y);
	} else if (t <= 7.5e+63) {
		tmp = t_1 * -(((y / x) - 0.5) * x);
	} else {
		tmp = (((t * t) * (sqrt((z * 2.0)) * 0.5)) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(2.0 * z))
	tmp = 0.0
	if (t <= 6.9e-6)
		tmp = Float64(t_1 * fma(0.5, x, Float64(-y)));
	elseif (t <= 7.5e+63)
		tmp = Float64(t_1 * Float64(-Float64(Float64(Float64(y / x) - 0.5) * x)));
	else
		tmp = Float64(Float64(Float64(Float64(t * t) * Float64(sqrt(Float64(z * 2.0)) * 0.5)) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t, 6.9e-6], N[(t$95$1 * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 7.5e+63], N[(t$95$1 * (-N[(N[(N[(y / x), $MachinePrecision] - 0.5), $MachinePrecision] * x), $MachinePrecision])), $MachinePrecision], N[(N[(N[(N[(t * t), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if t < 6.9e-6

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 71.0

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 71.0%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   if 6.9e-6 < t < 7.5000000000000005e63

   1. Initial program 99.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 17.9

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 17.9%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   6. Taylor expanded in x around -inf

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-x \cdot \left(\frac{y}{x} - \frac{1}{2}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      5. lower--.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - \frac{1}{2}\right) \cdot x\right) \]

      6. lower-/.f64 26.1

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   8. Applied rewrites 26.1%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-\left(\frac{y}{x} - 0.5\right) \cdot x\right) \]

   if 7.5000000000000005e63 < t

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right) + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right) \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}, {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      11. pow2 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      14. lift-*.f64 74.3

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t \cdot t, 0.5\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 74.3%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t \cdot t, 0.5\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   9. Taylor expanded in t around 0

      \[\leadsto \left(\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2} + \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       3. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       8. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       9. sqrt-prod N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       10. lift-sqrt.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       11. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       12. *-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2} \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       13. sqrt-prod N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       14. lift-sqrt.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       15. lift-*.f64 60.0

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, 0.5, \sqrt{2 \cdot z}\right) \cdot 0.5\right) \cdot x \]

   11. Applied rewrites 60.0%

       \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, 0.5, \sqrt{2 \cdot z}\right) \cdot 0.5\right) \cdot x \]

   12. Taylor expanded in t around inf

       \[\leadsto \left(\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       2. associate-*l* N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       3. sqrt-prod N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       6. associate-*l* N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       9. sqrt-prod N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       10. *-commutative N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       11. *-commutative N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       13. pow2 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       15. *-commutative N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       17. sqrt-prod N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       18. lift-sqrt.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       19. lift-*.f64 60.0

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x \]

   14. Applied rewrites 60.0%

       \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 66.0% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{+63}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 5.3e+63)
   (* (sqrt (* 2.0 z)) (fma 0.5 x (- y)))
   (* (* (* (* t t) (* (sqrt (* z 2.0)) 0.5)) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 5.3e+63) {
		tmp = sqrt((2.0 * z)) * fma(0.5, x, -y);
	} else {
		tmp = (((t * t) * (sqrt((z * 2.0)) * 0.5)) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 5.3e+63)
		tmp = Float64(sqrt(Float64(2.0 * z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(Float64(Float64(Float64(t * t) * Float64(sqrt(Float64(z * 2.0)) * 0.5)) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 5.3e+63], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(t * t), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 5.2999999999999999e63

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 67.5

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 67.5%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   if 5.2999999999999999e63 < t

   1. Initial program 99.3%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 74.3%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2} \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right)\right)} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} \cdot \left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right) + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right)\right) \cdot {t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      4. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left({t}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right) + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}\right) \cdot {t}^{2} + 1, {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2} + \frac{1}{6} \cdot {t}^{2}, {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      7. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6} \cdot {t}^{2} + \frac{1}{2}, {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, {t}^{2}, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      9. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      10. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), {t}^{2}, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      11. pow2 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      12. lift-*.f64 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), {t}^{2}, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      13. pow2 N/A

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, t \cdot t, \frac{1}{2}\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      14. lift-*.f64 74.3

          \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t \cdot t, 0.5\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 74.3%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, t \cdot t, 0.5\right), t \cdot t, 1\right), t \cdot t, 1\right)} \cdot 0.5\right) \cdot x \]

   9. Taylor expanded in t around 0

      \[\leadsto \left(\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) + \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2} + \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       2. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       3. associate-*l* N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left({t}^{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       6. pow2 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{z} \cdot \sqrt{2}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       8. *-commutative N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \left(\sqrt{2} \cdot \sqrt{z}\right), \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       9. sqrt-prod N/A

          \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       10. lift-sqrt.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       11. lift-*.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       12. *-commutative N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2} \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       13. sqrt-prod N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       14. lift-sqrt.f64 N/A

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, \frac{1}{2}, \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot x \]

       15. lift-*.f64 59.9

           \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, 0.5, \sqrt{2 \cdot z}\right) \cdot 0.5\right) \cdot x \]

   11. Applied rewrites 59.9%

       \[\leadsto \left(\mathsf{fma}\left(\left(t \cdot t\right) \cdot \sqrt{2 \cdot z}, 0.5, \sqrt{2 \cdot z}\right) \cdot 0.5\right) \cdot x \]

   12. Taylor expanded in t around inf

       \[\leadsto \left(\left(\frac{1}{2} \cdot \left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(\left(\left(\left({t}^{2} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       2. associate-*l* N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \left(\sqrt{2} \cdot \sqrt{z}\right)\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       3. sqrt-prod N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       5. lift-*.f64 N/A

          \[\leadsto \left(\left(\left({t}^{2} \cdot \sqrt{2 \cdot z}\right) \cdot \frac{1}{2}\right) \cdot \frac{1}{2}\right) \cdot x \]

       6. associate-*l* N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       7. lift-*.f64 N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       8. lift-sqrt.f64 N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       9. sqrt-prod N/A

          \[\leadsto \left(\left({t}^{2} \cdot \left(\left(\sqrt{2} \cdot \sqrt{z}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       10. *-commutative N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       11. *-commutative N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       12. lower-*.f64 N/A

           \[\leadsto \left(\left({t}^{2} \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       13. pow2 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       14. lift-*.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \sqrt{2}\right)\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       15. *-commutative N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       16. lower-*.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       17. sqrt-prod N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       18. lift-sqrt.f64 N/A

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot \frac{1}{2}\right)\right) \cdot \frac{1}{2}\right) \cdot x \]

       19. lift-*.f64 59.9

           \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x \]

   14. Applied rewrites 59.9%

       \[\leadsto \left(\left(\left(t \cdot t\right) \cdot \left(\sqrt{z \cdot 2} \cdot 0.5\right)\right) \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 86.2% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (- (* 0.5 x) y) (* (sqrt (* 2.0 z)) (fma (* t t) 0.5 1.0))))

C

double code(double x, double y, double z, double t) {
	return ((0.5 * x) - y) * (sqrt((2.0 * z)) * fma((t * t), 0.5, 1.0));
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * Float64(sqrt(Float64(2.0 * z)) * fma(Float64(t * t), 0.5, 1.0)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(0.5 * x), $MachinePrecision] - y), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   5. exp-sqrt N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   7. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

   8. exp-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   10. lower-exp.f64 99.4

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

4. Applied rewrites 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{{\left(e^{t}\right)}^{t}}} \]

5. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   5. lift-*.f64 85.6

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

7. Applied rewrites 85.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right)} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   3. lift--.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\sqrt{z \cdot 2}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   7. sqrt-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \color{blue}{\left(\sqrt{z} \cdot \sqrt{2}\right)}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   8. associate-*l* N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} - y\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \frac{1}{2} - y\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \]

   10. *-commutative N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \cdot \left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \color{blue}{\left(\left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \left(\color{blue}{\left(\sqrt{2} \cdot \sqrt{z}\right)} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   15. sqrt-prod N/A

       \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \left(\color{blue}{\sqrt{2 \cdot z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   16. lift-sqrt.f64 N/A

       \[\leadsto \left(\frac{1}{2} \cdot x - y\right) \cdot \left(\color{blue}{\sqrt{2 \cdot z}} \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right)\right) \]

   17. lift-*.f64 86.2

       \[\leadsto \left(0.5 \cdot x - y\right) \cdot \left(\sqrt{\color{blue}{2 \cdot z}} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right) \]

9. Applied rewrites 86.2%

   \[\leadsto \color{blue}{\left(0.5 \cdot x - y\right) \cdot \left(\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)} \]
10. Add Preprocessing

Alternative 12: 65.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3.4 \cdot 10^{+40}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3.4e+40)
   (* (sqrt (* 2.0 z)) (fma 0.5 x (- y)))
   (* (* (sqrt (* (* 2.0 z) (fma t t 1.0))) 0.5) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3.4e+40) {
		tmp = sqrt((2.0 * z)) * fma(0.5, x, -y);
	} else {
		tmp = (sqrt(((2.0 * z) * fma(t, t, 1.0))) * 0.5) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3.4e+40)
		tmp = Float64(sqrt(Float64(2.0 * z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * 0.5) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3.4e+40], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 3.39999999999999989e40

   1. Initial program 99.4%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 68.6

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 68.6%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   if 3.39999999999999989e40 < t

   1. Initial program 99.2%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(1 + {t}^{2}\right)} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left({t}^{2} + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      2. pow2 N/A

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \left(t \cdot t + 1\right)} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-fma.f64 55.7

         \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 55.7%

      \[\leadsto \left(\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot 0.5\right) \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 85.6% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z + z}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (+ z z))) (fma (* t t) 0.5 1.0)))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z + z))) * fma((t * t), 0.5, 1.0);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z + z))) * fma(Float64(t * t), 0.5, 1.0))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{e^{\frac{t \cdot t}{2}}} \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{t \cdot t}}{2}} \]

   3. lift-/.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\color{blue}{\frac{t \cdot t}{2}}} \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{\color{blue}{{t}^{2}}}{2}} \]

   5. exp-sqrt N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{e^{{t}^{2}}}} \]

   7. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{e^{\color{blue}{t \cdot t}}} \]

   8. exp-prod N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   9. lower-pow.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{\color{blue}{{\left(e^{t}\right)}^{t}}} \]

   10. lower-exp.f64 99.4

       \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \sqrt{{\color{blue}{\left(e^{t}\right)}}^{t}} \]

4. Applied rewrites 99.4%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\sqrt{{\left(e^{t}\right)}^{t}}} \]

5. Taylor expanded in t around 0

   \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + \frac{1}{2} \cdot {t}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left(\frac{1}{2} \cdot {t}^{2} + \color{blue}{1}\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \left({t}^{2} \cdot \frac{1}{2} + 1\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left({t}^{2}, \color{blue}{\frac{1}{2}}, 1\right) \]

   4. pow2 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   5. lift-*.f64 85.6

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

7. Applied rewrites 85.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\mathsf{fma}\left(t \cdot t, 0.5, 1\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z \cdot 2}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   2. *-commutative N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{2 \cdot z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   3. count-2-rev N/A

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, \frac{1}{2}, 1\right) \]

   4. lower-+.f64 85.6

      \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]

9. Applied rewrites 85.6%

   \[\leadsto \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{\color{blue}{z + z}}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right) \]
10. Add Preprocessing

Alternative 14: 43.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\sqrt{z + z} \cdot 0.5\right) \cdot x\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{2 \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* (sqrt (+ z z)) 0.5) x)))
   (if (<= x -6.2e+40)
     t_1
     (if (<= x 1.26e+133) (* (sqrt (* 2.0 z)) (- y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (sqrt((z + z)) * 0.5) * x;
	double tmp;
	if (x <= -6.2e+40) {
		tmp = t_1;
	} else if (x <= 1.26e+133) {
		tmp = sqrt((2.0 * z)) * -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (sqrt((z + z)) * 0.5d0) * x
    if (x <= (-6.2d+40)) then
        tmp = t_1
    else if (x <= 1.26d+133) then
        tmp = sqrt((2.0d0 * z)) * -y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (Math.sqrt((z + z)) * 0.5) * x;
	double tmp;
	if (x <= -6.2e+40) {
		tmp = t_1;
	} else if (x <= 1.26e+133) {
		tmp = Math.sqrt((2.0 * z)) * -y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (math.sqrt((z + z)) * 0.5) * x
	tmp = 0
	if x <= -6.2e+40:
		tmp = t_1
	elif x <= 1.26e+133:
		tmp = math.sqrt((2.0 * z)) * -y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(sqrt(Float64(z + z)) * 0.5) * x)
	tmp = 0.0
	if (x <= -6.2e+40)
		tmp = t_1;
	elseif (x <= 1.26e+133)
		tmp = Float64(sqrt(Float64(2.0 * z)) * Float64(-y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (sqrt((z + z)) * 0.5) * x;
	tmp = 0.0;
	if (x <= -6.2e+40)
		tmp = t_1;
	elseif (x <= 1.26e+133)
		tmp = sqrt((2.0 * z)) * -y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -6.2e+40], t$95$1, If[LessEqual[x, 1.26e+133], N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999995e40 or 1.2599999999999999e133 < x

   1. Initial program 99.8%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \color{blue}{\left(x \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \color{blue}{x}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \color{blue}{x}\right) \]

      4. associate-*l* N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(\frac{1}{2} \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \cdot \color{blue}{x} \]

   5. Applied rewrites 87.5%

      \[\leadsto \color{blue}{\left(\sqrt{\left(2 \cdot z\right) \cdot {\left(e^{t}\right)}^{t}} \cdot 0.5\right) \cdot x} \]

   6. Taylor expanded in t around 0

      \[\leadsto \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right) \cdot x \]
   7. Step-by-step derivation

      1. lift-*.f64 51.8

         \[\leadsto \left(\sqrt{2 \cdot z} \cdot 0.5\right) \cdot x \]

   8. Applied rewrites 51.8%

      \[\leadsto \left(\sqrt{2 \cdot z} \cdot 0.5\right) \cdot x \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(\sqrt{2 \cdot z} \cdot \frac{1}{2}\right) \cdot x \]

      2. count-2-rev N/A

         \[\leadsto \left(\sqrt{z + z} \cdot \frac{1}{2}\right) \cdot x \]

      3. lower-+.f64 51.8

         \[\leadsto \left(\sqrt{z + z} \cdot 0.5\right) \cdot x \]

   10. Applied rewrites 51.8%

       \[\leadsto \left(\sqrt{z + z} \cdot 0.5\right) \cdot x \]

   if -6.1999999999999995e40 < x < 1.2599999999999999e133

   1. Initial program 99.1%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

      2. sqrt-prod N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      8. *-commutative N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

      10. *-commutative N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

      11. *-lft-identity N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

      12. metadata-eval N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

      13. fp-cancel-sign-sub-inv N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

      15. mul-1-neg N/A

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

      16. lower-neg.f64 54.3

          \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

   5. Applied rewrites 54.3%

      \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

      2. lift-neg.f64 38.7

         \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

   8. Applied rewrites 38.7%

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 57.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (* 2.0 z)) (fma 0.5 x (- y))))

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * fma(0.5, x, -y);
}

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(2.0 * z)) * fma(0.5, x, Float64(-y)))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

   2. sqrt-prod N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

   11. *-lft-identity N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

   15. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

   16. lower-neg.f64 57.0

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

5. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]
6. Add Preprocessing

Alternative 16: 29.4% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot z} \cdot \left(-y\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* (sqrt (* 2.0 z)) (- y)))

C

double code(double x, double y, double z, double t) {
	return sqrt((2.0 * z)) * -y;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = sqrt((2.0d0 * z)) * -y
end function

Java

public static double code(double x, double y, double z, double t) {
	return Math.sqrt((2.0 * z)) * -y;
}

Python

def code(x, y, z, t):
	return math.sqrt((2.0 * z)) * -y

Julia

function code(x, y, z, t)
	return Float64(sqrt(Float64(2.0 * z)) * Float64(-y))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = sqrt((2.0 * z)) * -y;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[Sqrt[N[(2.0 * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]

Derivation

1. Initial program 99.3%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
2. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\sqrt{z} \cdot \left(\sqrt{2} \cdot \left(\frac{1}{2} \cdot x - y\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(\sqrt{z} \cdot \sqrt{2}\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x - y\right)} \]

   2. sqrt-prod N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{\frac{1}{2} \cdot x} - y\right) \]

   3. *-commutative N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(x \cdot \frac{1}{2} - y\right) \]

   4. lower-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \color{blue}{\left(x \cdot \frac{1}{2} - y\right)} \]

   5. lift-sqrt.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x \cdot \frac{1}{2}} - y\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \sqrt{z \cdot 2} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   8. *-commutative N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\color{blue}{x} \cdot \frac{1}{2} - y\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - y\right) \]

   11. *-lft-identity N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - 1 \cdot \color{blue}{y}\right) \]

   12. metadata-eval N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x - \left(\mathsf{neg}\left(-1\right)\right) \cdot y\right) \]

   13. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \left(\frac{1}{2} \cdot x + \color{blue}{-1 \cdot y}\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{x}, -1 \cdot y\right) \]

   15. mul-1-neg N/A

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(\frac{1}{2}, x, \mathsf{neg}\left(y\right)\right) \]

   16. lower-neg.f64 57.0

       \[\leadsto \sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right) \]

5. Applied rewrites 57.0%

   \[\leadsto \color{blue}{\sqrt{2 \cdot z} \cdot \mathsf{fma}\left(0.5, x, -y\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-1 \cdot \color{blue}{y}\right) \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(\mathsf{neg}\left(y\right)\right) \]

   2. lift-neg.f64 29.4

      \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]

8. Applied rewrites 29.4%

   \[\leadsto \sqrt{2 \cdot z} \cdot \left(-y\right) \]
9. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot {\left(e^{1}\right)}^{\left(\frac{t \cdot t}{2}\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (pow (exp 1.0) (/ (* t t) 2.0))))

C

double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * sqrt((z * 2.0))) * pow(exp(1.0), ((t * t) / 2.0));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((x * 0.5d0) - y) * sqrt((z * 2.0d0))) * (exp(1.0d0) ** ((t * t) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((x * 0.5) - y) * Math.sqrt((z * 2.0))) * Math.pow(Math.exp(1.0), ((t * t) / 2.0));
}

Python

def code(x, y, z, t):
	return (((x * 0.5) - y) * math.sqrt((z * 2.0))) * math.pow(math.exp(1.0), ((t * t) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * (exp(1.0) ^ Float64(Float64(t * t) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * (exp(1.0) ^ ((t * t) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[1.0], $MachinePrecision], N[(N[(t * t), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2025089 
(FPCore (x y z t)
  :name "Data.Number.Erf:$cinvnormcdf from erf-2.0.0.0, A"
  :precision binary64

  :alt
  (! :herbie-platform default (* (* (- (* x 1/2) y) (sqrt (* z 2))) (pow (exp 1) (/ (* t t) 2))))

  (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (exp (/ (* t t) 2.0))))