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

Percentage Accurate: 99.4% → 99.4%

Time: 6.1s

Alternatives: 13

Speedup: 1.0×

• 44.7% 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: 99.4% 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: 99.4% 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 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 98.7

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

4. Applied rewrites 98.7%

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

Alternative 2: 92.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z \cdot 2}\\ \mathbf{if}\;t \leq 50000000000 \lor \neg \left(t \leq 9 \cdot 10^{+65}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right) \cdot t, t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot t\_1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right) \cdot t, t, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (* z 2.0))))
   (if (or (<= t 50000000000.0) (not (<= t 9e+65)))
     (* (* (- (* x 0.5) y) t_1) (fma (* (fma 0.125 (* t t) 0.5) t) t 1.0))
     (*
      (* (- y) t_1)
      (fma
       (* (fma (fma (* t t) 0.020833333333333332 0.125) (* t t) 0.5) t)
       t
       1.0)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z * 2.0));
	double tmp;
	if ((t <= 50000000000.0) || !(t <= 9e+65)) {
		tmp = (((x * 0.5) - y) * t_1) * fma((fma(0.125, (t * t), 0.5) * t), t, 1.0);
	} else {
		tmp = (-y * t_1) * fma((fma(fma((t * t), 0.020833333333333332, 0.125), (t * t), 0.5) * t), t, 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z * 2.0))
	tmp = 0.0
	if ((t <= 50000000000.0) || !(t <= 9e+65))
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * t_1) * fma(Float64(fma(0.125, Float64(t * t), 0.5) * t), t, 1.0));
	else
		tmp = Float64(Float64(Float64(-y) * t_1) * fma(Float64(fma(fma(Float64(t * t), 0.020833333333333332, 0.125), Float64(t * t), 0.5) * t), t, 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t, 50000000000.0], N[Not[LessEqual[t, 9e+65]], $MachinePrecision]], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[(N[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[((-y) * t$95$1), $MachinePrecision] * N[(N[(N[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < 5e10 or 9e65 < t

   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

      \[\leadsto \left(\left(x \cdot \frac{1}{2} - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \color{blue}{\left(1 + {t}^{2} \cdot \left(\frac{1}{2} + {t}^{2} \cdot \left(\frac{1}{8} + \frac{1}{48} \cdot {t}^{2}\right)\right)\right)} \]
   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 94.2

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

   5. Applied rewrites 94.2%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 94.2%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 91.0%

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

   if 5e10 < t < 9e65

   1. Initial program 92.9%

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

   3. Taylor expanded in 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 79.1

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

   5. Applied rewrites 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-*r* N/A

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

      8. lower-fma.f64 N/A

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

   7. Applied rewrites 79.1%

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

   8. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{\left(-1 \cdot y\right)} \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, \frac{1}{48}, \frac{1}{8}\right), t \cdot t, \frac{1}{2}\right) \cdot t, t, 1\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lift-neg.f64 64.9

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

   10. Applied rewrites 64.9%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 50000000000 \lor \neg \left(t \leq 9 \cdot 10^{+65}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.125, t \cdot t, 0.5\right) \cdot t, t, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(t \cdot t, 0.020833333333333332, 0.125\right), t \cdot t, 0.5\right) \cdot t, t, 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(0.5 * x) - y) * Float64(sqrt(Float64(2.0 * z)) * fma(fma(fma(Float64(t * t), 0.020833333333333332, 0.125), Float64(t * t), 0.5), Float64(t * t), 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[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * N[(t * t), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 93.3

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

5. Applied rewrites 93.3%

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

7. Applied rewrites 94.9%

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

Alternative 4: 94.8% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (*
  (* (- (* x 0.5) y) (sqrt (* z 2.0)))
  (fma
   (* (fma (fma (* t t) 0.020833333333333332 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((t * t), 0.020833333333333332, 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(Float64(fma(fma(Float64(t * t), 0.020833333333333332, 0.125), Float64(t * t), 0.5) * 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[(N[(N[(t * t), $MachinePrecision] * 0.020833333333333332 + 0.125), $MachinePrecision] * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 93.3

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

5. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 93.3%

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

8. Final simplification 93.3%

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

Alternative 5: 86.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+85} \lor \neg \left(t \leq 2.25 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t 4e+85) (not (<= t 2.25e+154)))
   (* (* (- (* x 0.5) y) (sqrt (* z 2.0))) (fma (* t t) 0.5 1.0))
   (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= 4e+85) || !(t <= 2.25e+154)) {
		tmp = (((x * 0.5) - y) * sqrt((z * 2.0))) * fma((t * t), 0.5, 1.0);
	} else {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= 4e+85) || !(t <= 2.25e+154))
		tmp = Float64(Float64(Float64(Float64(x * 0.5) - y) * sqrt(Float64(z * 2.0))) * fma(Float64(t * t), 0.5, 1.0));
	else
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, 4e+85], N[Not[LessEqual[t, 2.25e+154]], $MachinePrecision]], N[(N[(N[(N[(x * 0.5), $MachinePrecision] - y), $MachinePrecision] * N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 4.0000000000000001e85 or 2.25000000000000005e154 < t

   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

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

      1. +-commutative N/A

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

      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 84.0

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

   5. Applied rewrites 84.0%

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

   if 4.0000000000000001e85 < t < 2.25000000000000005e154

   1. Initial program 90.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 90.3

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

   8. Applied rewrites 90.3%

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

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\left(-\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z}\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4 \cdot 10^{+85} \lor \neg \left(t \leq 2.25 \cdot 10^{+154}\right):\\ \;\;\;\;\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 92.4% accurate, 2.7× speedup

Math

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

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(Float64(fma(0.125, Float64(t * t), 0.5) * 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[(0.125 * N[(t * t), $MachinePrecision] + 0.5), $MachinePrecision] * t), $MachinePrecision] * t + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 93.3

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

5. Applied rewrites 93.3%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. lower-fma.f64 N/A

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

7. Applied rewrites 93.3%

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

8. Taylor expanded in t around 0

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

10. Applied rewrites 88.1%

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

11. Final simplification 88.1%

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

Alternative 7: 68.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3e-8)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (if (<= t 2.4e+219)
     (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))
     (* (* 0.5 x) (* (sqrt (* z 2.0)) (fma (* t t) 0.5 1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-8) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else if (t <= 2.4e+219) {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	} else {
		tmp = (0.5 * x) * (sqrt((z * 2.0)) * fma((t * t), 0.5, 1.0));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3e-8)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	elseif (t <= 2.4e+219)
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	else
		tmp = Float64(Float64(0.5 * x) * Float64(sqrt(Float64(z * 2.0)) * fma(Float64(t * t), 0.5, 1.0)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3e-8], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.4e+219], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision], N[(N[(0.5 * x), $MachinePrecision] * N[(N[Sqrt[N[(z * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] * 0.5 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < 2.99999999999999973e-8

   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 67.3

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

   5. Applied rewrites 67.3%

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

   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 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.2

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

   10. Applied rewrites 35.2%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-neg.f64 67.3

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

   13. Applied rewrites 67.3%

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

   if 2.99999999999999973e-8 < t < 2.4e219

   1. Initial program 95.8%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 81.2%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 63.4

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

   8. Applied rewrites 63.4%

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

   10. Applied rewrites 65.4%

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

   if 2.4e219 < t

   1. Initial program 100.0%

      \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
   2. Add Preprocessing
   3. 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 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 84.2

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

   7. Applied rewrites 84.2%

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

   8. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 84.2

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

   10. Applied rewrites 84.2%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-sqrt.f64 N/A

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

       5. sqrt-prod N/A

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

       6. associate-*l* N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. sqrt-prod N/A

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

       10. lift-sqrt.f64 N/A

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

       11. lift-*.f64 89.5

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

   12. Applied rewrites 89.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{+219}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;\left(0.5 \cdot x\right) \cdot \left(\sqrt{z \cdot 2} \cdot \mathsf{fma}\left(t \cdot t, 0.5, 1\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3e-8)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (sqrt (* (fma (fma t t 2.0) (* t t) 2.0) z)) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-8) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = sqrt((fma(fma(t, t, 2.0), (t * t), 2.0) * z)) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3e-8)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(sqrt(Float64(fma(fma(t, t, 2.0), Float64(t * t), 2.0) * z)) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3e-8], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(t * t + 2.0), $MachinePrecision] * N[(t * t), $MachinePrecision] + 2.0), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999973e-8

   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 67.3

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

   5. Applied rewrites 67.3%

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

   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 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.2

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

   10. Applied rewrites 35.2%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-neg.f64 67.3

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

   13. Applied rewrites 67.3%

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

   if 2.99999999999999973e-8 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. distribute-rgt-out N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-*.f64 64.8

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

   8. Applied rewrites 64.8%

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

   10. Applied rewrites 66.2%

       \[\leadsto \color{blue}{\left(-\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z}\right) \cdot y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(t, t, 2\right), t \cdot t, 2\right) \cdot z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.3% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z + \left(t \cdot t\right) \cdot z\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3e-8)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (sqrt (* 2.0 (+ z (* (* t t) z)))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-8) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = sqrt((2.0 * (z + ((t * t) * z)))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3e-8)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(z + Float64(Float64(t * t) * z)))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3e-8], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(z + N[(N[(t * t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999973e-8

   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 67.3

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

   5. Applied rewrites 67.3%

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

   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 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.2

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

   10. Applied rewrites 35.2%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-neg.f64 67.3

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

   13. Applied rewrites 67.3%

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

   if 2.99999999999999973e-8 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around 0

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

      1. distribute-lft-out N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

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

      6. lift-*.f64 50.5

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

   8. Applied rewrites 50.5%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(z + \left(t \cdot t\right) \cdot z\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 3e-8)
   (* (sqrt (+ z z)) (fma 0.5 x (- y)))
   (* (sqrt (* (* 2.0 z) (fma t t 1.0))) (- y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 3e-8) {
		tmp = sqrt((z + z)) * fma(0.5, x, -y);
	} else {
		tmp = sqrt(((2.0 * z) * fma(t, t, 1.0))) * -y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 3e-8)
		tmp = Float64(sqrt(Float64(z + z)) * fma(0.5, x, Float64(-y)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * z) * fma(t, t, 1.0))) * Float64(-y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 3e-8], N[(N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision] * N[(0.5 * x + (-y)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * z), $MachinePrecision] * N[(t * t + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-y)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 2.99999999999999973e-8

   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 67.3

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

   5. Applied rewrites 67.3%

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

   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 35.2

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

   8. Applied rewrites 35.2%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 35.2

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

   10. Applied rewrites 35.2%

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

   11. Taylor expanded in y around 0

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

       1. mul-1-neg N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lift-neg.f64 67.3

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

   13. Applied rewrites 67.3%

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

   if 2.99999999999999973e-8 < t

   1. Initial program 97.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(y \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right) \cdot \sqrt{z}\right) \]

      2. associate-*l* N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right) \cdot \sqrt{z}\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(y \cdot \left(\sqrt{z} \cdot \left(e^{\frac{1}{2} \cdot {t}^{2}} \cdot \sqrt{2}\right)\right)\right) \]

      4. distribute-rgt-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 77.6%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. pow2 N/A

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

      3. lower-fma.f64 50.5

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

   8. Applied rewrites 50.5%

      \[\leadsto \left(-\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)}\right) \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 3 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{z + z} \cdot \mathsf{fma}\left(0.5, x, -y\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(2 \cdot z\right) \cdot \mathsf{fma}\left(t, t, 1\right)} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \sqrt{z + z}\\ \mathbf{if}\;x \leq -8 \cdot 10^{+50} \lor \neg \left(x \leq 1.4 \cdot 10^{-38}\right):\\ \;\;\;\;t\_1 \cdot \left(0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \left(-y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (sqrt (+ z z))))
   (if (or (<= x -8e+50) (not (<= x 1.4e-38)))
     (* t_1 (* 0.5 x))
     (* t_1 (- y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = sqrt((z + z));
	double tmp;
	if ((x <= -8e+50) || !(x <= 1.4e-38)) {
		tmp = t_1 * (0.5 * x);
	} else {
		tmp = t_1 * -y;
	}
	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))
    if ((x <= (-8d+50)) .or. (.not. (x <= 1.4d-38))) then
        tmp = t_1 * (0.5d0 * x)
    else
        tmp = t_1 * -y
    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));
	double tmp;
	if ((x <= -8e+50) || !(x <= 1.4e-38)) {
		tmp = t_1 * (0.5 * x);
	} else {
		tmp = t_1 * -y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.sqrt((z + z))
	tmp = 0
	if (x <= -8e+50) or not (x <= 1.4e-38):
		tmp = t_1 * (0.5 * x)
	else:
		tmp = t_1 * -y
	return tmp

Julia

function code(x, y, z, t)
	t_1 = sqrt(Float64(z + z))
	tmp = 0.0
	if ((x <= -8e+50) || !(x <= 1.4e-38))
		tmp = Float64(t_1 * Float64(0.5 * x));
	else
		tmp = Float64(t_1 * Float64(-y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = sqrt((z + z));
	tmp = 0.0;
	if ((x <= -8e+50) || ~((x <= 1.4e-38)))
		tmp = t_1 * (0.5 * x);
	else
		tmp = t_1 * -y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[Sqrt[N[(z + z), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -8e+50], N[Not[LessEqual[x, 1.4e-38]], $MachinePrecision]], N[(t$95$1 * N[(0.5 * x), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * (-y)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.0000000000000006e50 or 1.4e-38 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in 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 12.6

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

   8. Applied rewrites 12.6%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 12.6

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

   10. Applied rewrites 12.6%

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

   11. Taylor expanded in x around inf

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

       1. lift-*.f64 46.5

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

   13. Applied rewrites 46.5%

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

   if -8.0000000000000006e50 < x < 1.4e-38

   1. Initial program 97.8%

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

   3. Taylor expanded in 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 51.2

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

   5. Applied rewrites 51.2%

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

   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 41.4

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

   8. Applied rewrites 41.4%

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

      1. lift-*.f64 N/A

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

      2. count-2-rev N/A

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

      3. lower-+.f64 41.4

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

   10. Applied rewrites 41.4%

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

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+50} \lor \neg \left(x \leq 1.4 \cdot 10^{-38}\right):\\ \;\;\;\;\sqrt{z + z} \cdot \left(0.5 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{z + z} \cdot \left(-y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 56.6% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 53.7

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

5. Applied rewrites 53.7%

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

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 29.0

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

8. Applied rewrites 29.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 29.0

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

10. Applied rewrites 29.0%

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

11. Taylor expanded in y around 0

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

    1. mul-1-neg N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lift-neg.f64 53.7

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

13. Applied rewrites 53.7%

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

Alternative 13: 29.4% accurate, 7.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return sqrt((z + 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((z + z)) * -y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.7%

   \[\left(\left(x \cdot 0.5 - y\right) \cdot \sqrt{z \cdot 2}\right) \cdot e^{\frac{t \cdot t}{2}} \]
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 53.7

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

5. Applied rewrites 53.7%

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

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 29.0

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

8. Applied rewrites 29.0%

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

   1. lift-*.f64 N/A

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

   2. count-2-rev N/A

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

   3. lower-+.f64 29.0

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

10. Applied rewrites 29.0%

    \[\leadsto \sqrt{z + z} \cdot \left(-y\right) \]
11. Add Preprocessing

Developer Target 1: 99.4% 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 2025051 
(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))))