Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Percentage Accurate: 86.5% → 99.5%

Time: 3.9s

Alternatives: 14

Speedup: 1.0×

• 76.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

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 / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

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

Initial Program: 86.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

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 / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y z t)
  :precision binary64
  (fma (fma (- 1.0 t) 2.0 (/ 2.0 z)) (/ 1.0 t) (/ x y)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 86.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. mult-flip N/A

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

   5. associate-*r/ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. times-frac N/A

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

   9. lower-fma.f64 N/A

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

3. Applied rewrites 99.5%

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

Alternative 2: 95.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}, y, x\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 50:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, \frac{1}{z}, \frac{x}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= (/ x y) -2.0)
  (/ (fma (/ (fma (- 1.0 t) 2.0 (/ 2.0 z)) t) y x) y)
  (if (<= (/ x y) 50.0)
    (fma 2.0 (/ (- 1.0 t) t) (* 2.0 (/ 1.0 (* t z))))
    (fma (/ 2.0 t) (/ 1.0 z) (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2.0) {
		tmp = fma((fma((1.0 - t), 2.0, (2.0 / z)) / t), y, x) / y;
	} else if ((x / y) <= 50.0) {
		tmp = fma(2.0, ((1.0 - t) / t), (2.0 * (1.0 / (t * z))));
	} else {
		tmp = fma((2.0 / t), (1.0 / z), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2.0)
		tmp = Float64(fma(Float64(fma(Float64(1.0 - t), 2.0, Float64(2.0 / z)) / t), y, x) / y);
	elseif (Float64(x / y) <= 50.0)
		tmp = fma(2.0, Float64(Float64(1.0 - t) / t), Float64(2.0 * Float64(1.0 / Float64(t * z))));
	else
		tmp = fma(Float64(2.0 / t), Float64(1.0 / z), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2.0], N[(N[(N[(N[(N[(1.0 - t), $MachinePrecision] * 2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 50.0], N[(2.0 * N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] + N[(2.0 * N[(1.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -2

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. add-to-fraction N/A

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

      5. lower-/.f64 N/A

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

   3. Applied rewrites 87.7%

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

   if -2 < (/.f64 x y) < 50

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

   if 50 < (/.f64 x y)

   1. Initial program 86.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} + \frac{x}{y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} + \frac{x}{y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{z}} + \frac{x}{y} \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{1}{z}} + \frac{x}{y} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.4%

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

   6. Applied rewrites 63.4%

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

Alternative 3: 92.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{1 - t}{t}\\ \mathbf{if}\;\frac{x}{y} \leq -100000000000:\\ \;\;\;\;\mathsf{fma}\left(2, t\_1, \frac{x}{y}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 50:\\ \;\;\;\;\mathsf{fma}\left(2, t\_1, 2 \cdot \frac{1}{t \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, \frac{1}{z}, \frac{x}{y}\right)\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ (- 1.0 t) t)))
  (if (<= (/ x y) -100000000000.0)
    (fma 2.0 t_1 (/ x y))
    (if (<= (/ x y) 50.0)
      (fma 2.0 t_1 (* 2.0 (/ 1.0 (* t z))))
      (fma (/ 2.0 t) (/ 1.0 z) (/ x y))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - t) / t;
	double tmp;
	if ((x / y) <= -100000000000.0) {
		tmp = fma(2.0, t_1, (x / y));
	} else if ((x / y) <= 50.0) {
		tmp = fma(2.0, t_1, (2.0 * (1.0 / (t * z))));
	} else {
		tmp = fma((2.0 / t), (1.0 / z), (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - t) / t)
	tmp = 0.0
	if (Float64(x / y) <= -100000000000.0)
		tmp = fma(2.0, t_1, Float64(x / y));
	elseif (Float64(x / y) <= 50.0)
		tmp = fma(2.0, t_1, Float64(2.0 * Float64(1.0 / Float64(t * z))));
	else
		tmp = fma(Float64(2.0 / t), Float64(1.0 / z), Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -100000000000.0], N[(2.0 * t$95$1 + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 50.0], N[(2.0 * t$95$1 + N[(2.0 * N[(1.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1e11

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   if -1e11 < (/.f64 x y) < 50

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

   if 50 < (/.f64 x y)

   1. Initial program 86.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} + \frac{x}{y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} + \frac{x}{y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{z}} + \frac{x}{y} \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{1}{z}} + \frac{x}{y} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.4%

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

   6. Applied rewrites 63.4%

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

Alternative 4: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, \frac{1 - t}{t}, \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.078:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{t}, \frac{1}{z}, \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fma 2.0 (/ (- 1.0 t) t) (/ x y))))
  (if (<= z -8.2e-12)
    t_1
    (if (<= z 0.078) (fma (/ 2.0 t) (/ 1.0 z) (/ x y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, ((1.0 - t) / t), (x / y));
	double tmp;
	if (z <= -8.2e-12) {
		tmp = t_1;
	} else if (z <= 0.078) {
		tmp = fma((2.0 / t), (1.0 / z), (x / y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(2.0, Float64(Float64(1.0 - t) / t), Float64(x / y))
	tmp = 0.0
	if (z <= -8.2e-12)
		tmp = t_1;
	elseif (z <= 0.078)
		tmp = fma(Float64(2.0 / t), Float64(1.0 / z), Float64(x / y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e-12], t$95$1, If[LessEqual[z, 0.078], N[(N[(2.0 / t), $MachinePrecision] * N[(1.0 / z), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999998e-12 or 0.078 < z

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   if -8.1999999999999998e-12 < z < 0.078

   1. Initial program 86.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} + \frac{x}{y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z}} + \frac{x}{y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      5. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t}}{z}} + \frac{x}{y} \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\frac{2}{t} \cdot \frac{1}{z}} + \frac{x}{y} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 63.4%

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

   6. Applied rewrites 63.4%

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

Alternative 5: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_4 := \mathsf{fma}\left(2, \frac{1}{t}, \frac{x}{y}\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2.5 \cdot 10^{+60}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -50000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq -2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+141}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{2 + 2 \cdot z}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
       (t_2 (+ (/ x y) -2.0))
       (t_3 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
       (t_4 (fma 2.0 (/ 1.0 t) (/ x y))))
  (if (<= t_3 -5e+168)
    t_1
    (if (<= t_3 -2.5e+60)
      t_4
      (if (<= t_3 -50000000.0)
        t_1
        (if (<= t_3 -2.0)
          t_2
          (if (<= t_3 2e+141)
            t_4
            (if (<= t_3 INFINITY)
              (/ (+ 2.0 (* 2.0 z)) (* t z))
              t_2))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_4 = fma(2.0, (1.0 / t), (x / y));
	double tmp;
	if (t_3 <= -5e+168) {
		tmp = t_1;
	} else if (t_3 <= -2.5e+60) {
		tmp = t_4;
	} else if (t_3 <= -50000000.0) {
		tmp = t_1;
	} else if (t_3 <= -2.0) {
		tmp = t_2;
	} else if (t_3 <= 2e+141) {
		tmp = t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (2.0 + (2.0 * z)) / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_4 = fma(2.0, Float64(1.0 / t), Float64(x / y))
	tmp = 0.0
	if (t_3 <= -5e+168)
		tmp = t_1;
	elseif (t_3 <= -2.5e+60)
		tmp = t_4;
	elseif (t_3 <= -50000000.0)
		tmp = t_1;
	elseif (t_3 <= -2.0)
		tmp = t_2;
	elseif (t_3 <= 2e+141)
		tmp = t_4;
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(2.0 + Float64(2.0 * z)) / Float64(t * z));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(1.0 / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+168], t$95$1, If[LessEqual[t$95$3, -2.5e+60], t$95$4, If[LessEqual[t$95$3, -50000000.0], t$95$1, If[LessEqual[t$95$3, -2.0], t$95$2, If[LessEqual[t$95$3, 2e+141], t$95$4, If[LessEqual[t$95$3, Infinity], N[(N[(2.0 + N[(2.0 * z), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999997e168 or -2.4999999999999999e60 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e7

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.4%

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} + 2}{t} \]

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      9. metadata-eval 48.4%

         \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   6. Applied rewrites 48.4%

      \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   if -4.9999999999999997e168 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2.4999999999999999e60 or -2 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2e141

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 52.0%

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

   if -5e7 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.6%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]

   if 2e141 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + 2 \cdot \color{blue}{\frac{1}{t \cdot z}} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{\color{blue}{t \cdot z}} \]

      4. mult-flip-rev N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{2}{\color{blue}{t \cdot z}} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{2}{t \cdot z} \]

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-add N/A

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

      13. lift-/.f64 N/A

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

      14. add-to-fraction N/A

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

      15. associate-/l/ N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2 N/A

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

      21. lift-+.f64 N/A

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

      22. *-commutative N/A

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2 + 2 \cdot z}{\color{blue}{t \cdot z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot \color{blue}{z}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

      4. lower-*.f64 48.3%

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

   9. Applied rewrites 48.3%

      \[\leadsto \frac{2 + 2 \cdot z}{\color{blue}{t \cdot z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(2, \frac{1 - t}{t}, \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.078:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (fma 2.0 (/ (- 1.0 t) t) (/ x y))))
  (if (<= z -8.2e-12)
    t_1
    (if (<= z 0.078) (+ (/ x y) (/ 2.0 (* t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(2.0, ((1.0 - t) / t), (x / y));
	double tmp;
	if (z <= -8.2e-12) {
		tmp = t_1;
	} else if (z <= 0.078) {
		tmp = (x / y) + (2.0 / (t * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(2.0, Float64(Float64(1.0 - t) / t), Float64(x / y))
	tmp = 0.0
	if (z <= -8.2e-12)
		tmp = t_1;
	elseif (z <= 0.078)
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 * N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -8.2e-12], t$95$1, If[LessEqual[z, 0.078], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999998e-12 or 0.078 < z

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   if -8.1999999999999998e-12 < z < 0.078

   1. Initial program 86.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 82.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -100000000000:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{t}, \frac{x}{y}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ 2.0 (* t z))))
  (if (<= (/ x y) -100000000000.0)
    (fma 2.0 (/ 1.0 t) (/ x y))
    (if (<= (/ x y) 5e-5) (fma -1.0 2.0 t_1) (+ (/ x y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double tmp;
	if ((x / y) <= -100000000000.0) {
		tmp = fma(2.0, (1.0 / t), (x / y));
	} else if ((x / y) <= 5e-5) {
		tmp = fma(-1.0, 2.0, t_1);
	} else {
		tmp = (x / y) + t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	tmp = 0.0
	if (Float64(x / y) <= -100000000000.0)
		tmp = fma(2.0, Float64(1.0 / t), Float64(x / y));
	elseif (Float64(x / y) <= 5e-5)
		tmp = fma(-1.0, 2.0, t_1);
	else
		tmp = Float64(Float64(x / y) + t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -100000000000.0], N[(2.0 * N[(1.0 / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e-5], N[(-1.0 * 2.0 + t$95$1), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1e11

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in t around 0

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

   7. Applied rewrites 52.0%

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

   if -1e11 < (/.f64 x y) < 5.0000000000000002e-5

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 49.6%

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

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot -1 + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto -1 \cdot 2 + \color{blue}{2} \cdot \frac{1}{t \cdot z} \]

      3. lower-fma.f64 49.6%

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip-rev N/A

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

      7. lift-/.f64 49.6%

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

   9. Applied rewrites 49.6%

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

   if 5.0000000000000002e-5 < (/.f64 x y)

   1. Initial program 86.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.0%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 82.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_1 \leq -50000000:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{elif}\;t\_1 \leq 10^{+133}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{2 + 2 \cdot z}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
       (t_2 (+ (/ x y) -2.0)))
  (if (<= t_1 -50000000.0)
    (/ (- (/ 2.0 z) -2.0) t)
    (if (<= t_1 1e+133)
      t_2
      (if (<= t_1 INFINITY) (/ (+ 2.0 (* 2.0 z)) (* t z)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) + -2.0;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else if (t_1 <= 1e+133) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (2.0 + (2.0 * z)) / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) + -2.0;
	double tmp;
	if (t_1 <= -50000000.0) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else if (t_1 <= 1e+133) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 + (2.0 * z)) / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	t_2 = (x / y) + -2.0
	tmp = 0
	if t_1 <= -50000000.0:
		tmp = ((2.0 / z) - -2.0) / t
	elif t_1 <= 1e+133:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (2.0 + (2.0 * z)) / (t * z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_2 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_1 <= -50000000.0)
		tmp = Float64(Float64(Float64(2.0 / z) - -2.0) / t);
	elseif (t_1 <= 1e+133)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(2.0 + Float64(2.0 * z)) / Float64(t * z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	t_2 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_1 <= -50000000.0)
		tmp = ((2.0 / z) - -2.0) / t;
	elseif (t_1 <= 1e+133)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (2.0 + (2.0 * z)) / (t * z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -50000000.0], N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 1e+133], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(2.0 + N[(2.0 * z), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e7

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.4%

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} + 2}{t} \]

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      9. metadata-eval 48.4%

         \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   6. Applied rewrites 48.4%

      \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   if -5e7 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1e133 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.6%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]

   if 1e133 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + 2 \cdot \color{blue}{\frac{1}{t \cdot z}} \]

      3. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{\color{blue}{t \cdot z}} \]

      4. mult-flip-rev N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{2}{\color{blue}{t \cdot z}} \]

      5. lift-/.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{2}{t \cdot z} \]

      6. associate-*r/ N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-add N/A

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

      13. lift-/.f64 N/A

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

      14. add-to-fraction N/A

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

      15. associate-/l/ N/A

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

      16. associate-*l* N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f64 N/A

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

      19. *-commutative N/A

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

      20. count-2 N/A

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

      21. lift-+.f64 N/A

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

      22. *-commutative N/A

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

   6. Applied rewrites 53.4%

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

   7. Taylor expanded in t around 0

      \[\leadsto \frac{2 + 2 \cdot z}{\color{blue}{t \cdot z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot \color{blue}{z}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

      4. lower-*.f64 48.3%

         \[\leadsto \frac{2 + 2 \cdot z}{t \cdot z} \]

   9. Applied rewrites 48.3%

      \[\leadsto \frac{2 + 2 \cdot z}{\color{blue}{t \cdot z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_2 \leq -50000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+133}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
       (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
       (t_3 (+ (/ x y) -2.0)))
  (if (<= t_2 -50000000.0)
    t_1
    (if (<= t_2 1e+133) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+133) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) + -2.0;
	double tmp;
	if (t_2 <= -50000000.0) {
		tmp = t_1;
	} else if (t_2 <= 1e+133) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((2.0 / z) - -2.0) / t
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	t_3 = (x / y) + -2.0
	tmp = 0
	if t_2 <= -50000000.0:
		tmp = t_1
	elif t_2 <= 1e+133:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_3 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_2 <= -50000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+133)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((2.0 / z) - -2.0) / t;
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	t_3 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_2 <= -50000000.0)
		tmp = t_1;
	elseif (t_2 <= 1e+133)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -50000000.0], t$95$1, If[LessEqual[t$95$2, 1e+133], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e7 or 1e133 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

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

   2. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.4%

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   4. Applied rewrites 48.4%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} + 2}{t} \]

      3. add-flip N/A

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

      4. lower--.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      9. metadata-eval 48.4%

         \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   6. Applied rewrites 48.4%

      \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   if -5e7 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1e133 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.6%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
       (t_2 (+ (/ x y) -2.0)))
  (if (<= t_1 -5e+168)
    (/ (/ 2.0 t) z)
    (if (<= t_1 5e+148)
      t_2
      (if (<= t_1 INFINITY) (/ 2.0 (* t z)) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) + -2.0;
	double tmp;
	if (t_1 <= -5e+168) {
		tmp = (2.0 / t) / z;
	} else if (t_1 <= 5e+148) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) + -2.0;
	double tmp;
	if (t_1 <= -5e+168) {
		tmp = (2.0 / t) / z;
	} else if (t_1 <= 5e+148) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	t_2 = (x / y) + -2.0
	tmp = 0
	if t_1 <= -5e+168:
		tmp = (2.0 / t) / z
	elif t_1 <= 5e+148:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = 2.0 / (t * z)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_2 = Float64(Float64(x / y) + -2.0)
	tmp = 0.0
	if (t_1 <= -5e+168)
		tmp = Float64(Float64(2.0 / t) / z);
	elseif (t_1 <= 5e+148)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(2.0 / Float64(t * z));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	t_2 = (x / y) + -2.0;
	tmp = 0.0;
	if (t_1 <= -5e+168)
		tmp = (2.0 / t) / z;
	elseif (t_1 <= 5e+148)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = 2.0 / (t * z);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+168], N[(N[(2.0 / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 5e+148], t$95$2, If[LessEqual[t$95$1, Infinity], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999997e168

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 31.5%

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   6. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      5. lower-/.f64 31.5%

         \[\leadsto \frac{\frac{2}{t}}{z} \]

   8. Applied rewrites 31.5%

      \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

   if -4.9999999999999997e168 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5.0000000000000002e148 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.6%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]

   if 5.0000000000000002e148 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 31.5%

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   6. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} + -2\\ t_3 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+148}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (let* ((t_1 (/ 2.0 (* t z)))
       (t_2 (+ (/ x y) -2.0))
       (t_3 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
  (if (<= t_3 -5e+168)
    t_1
    (if (<= t_3 5e+148) t_2 (if (<= t_3 INFINITY) t_1 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+168) {
		tmp = t_1;
	} else if (t_3 <= 5e+148) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (x / y) + -2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+168) {
		tmp = t_1;
	} else if (t_3 <= 5e+148) {
		tmp = t_2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (t * z)
	t_2 = (x / y) + -2.0
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if t_3 <= -5e+168:
		tmp = t_1
	elif t_3 <= 5e+148:
		tmp = t_2
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	t_2 = Float64(Float64(x / y) + -2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_3 <= -5e+168)
		tmp = t_1;
	elseif (t_3 <= 5e+148)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (t * z);
	t_2 = (x / y) + -2.0;
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if (t_3 <= -5e+168)
		tmp = t_1;
	elseif (t_3 <= 5e+148)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+168], t$95$1, If[LessEqual[t$95$3, 5e+148], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -4.9999999999999997e168 or 5.0000000000000002e148 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. times-frac N/A

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

      9. lower-fma.f64 N/A

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 31.5%

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   6. Applied rewrites 31.5%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

   if -4.9999999999999997e168 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5.0000000000000002e148 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

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

   2. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.6%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 53.6% accurate, 3.4× speedup

Math

\[\frac{x}{y} + -2 \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (+ (/ x y) -2.0))

C

double code(double x, double y, double z, double t) {
	return (x / y) + -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 / y) + (-2.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + -2.0;
}

Python

def code(x, y, z, t):
	return (x / y) + -2.0

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + -2.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + -2.0;
end

Wolfram

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

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in t around inf

   \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
3. Step-by-step derivation

4. Applied rewrites 53.6%

   \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
5. Add Preprocessing

Alternative 13: 36.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -0.0004:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

FPCore

(FPCore (x y z t)
  :precision binary64
  (if (<= t -0.0004) -2.0 (if (<= t 1.75e-13) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0004) {
		tmp = -2.0;
	} else if (t <= 1.75e-13) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	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) :: tmp
    if (t <= (-0.0004d0)) then
        tmp = -2.0d0
    else if (t <= 1.75d-13) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.0004) {
		tmp = -2.0;
	} else if (t <= 1.75e-13) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -0.0004:
		tmp = -2.0
	elif t <= 1.75e-13:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -0.0004)
		tmp = -2.0;
	elseif (t <= 1.75e-13)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -0.0004)
		tmp = -2.0;
	elseif (t <= 1.75e-13)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -0.0004], -2.0, If[LessEqual[t, 1.75e-13], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -4.0000000000000002e-4 or 1.7500000000000001e-13 < t

   1. Initial program 86.5%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 66.9%

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

   4. Applied rewrites 66.9%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in t around inf

      \[\leadsto -2 \]
   9. Step-by-step derivation

   10. Applied rewrites 20.4%

       \[\leadsto -2 \]

   if -4.0000000000000002e-4 < t < 1.7500000000000001e-13

   1. Initial program 86.5%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 70.7%

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

   4. Applied rewrites 70.7%

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

   5. Taylor expanded in t around 0

      \[\leadsto \frac{2}{\color{blue}{t}} \]
   6. Step-by-step derivation

      1. lower-/.f64 19.2%

         \[\leadsto \frac{2}{t} \]

   7. Applied rewrites 19.2%

      \[\leadsto \frac{2}{\color{blue}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 20.4% accurate, 25.4× speedup

Math

\[-2 \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -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 = -2.0d0
end function

Java

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

Python

def code(x, y, z, t):
	return -2.0

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := -2.0

Derivation

1. Initial program 86.5%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
3. Step-by-step derivation

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 66.9%

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

4. Applied rewrites 66.9%

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

5. Taylor expanded in t around inf

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

7. Applied rewrites 49.6%

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

8. Taylor expanded in t around inf

   \[\leadsto -2 \]
9. Step-by-step derivation

10. Applied rewrites 20.4%

    \[\leadsto -2 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64
  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))