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

Percentage Accurate: 86.9% → 99.2%

Time: 4.2s

Alternatives: 12

Speedup: 1.3×

• 23.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 86.9% accurate, 1.0× speedup

Math

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

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: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{x}{y} + \left(\frac{\frac{2}{z} - -2}{t} - 2\right) \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 86.9%

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

2. Taylor expanded in t around inf

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

   1. lower--.f64 N/A

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

   2. associate-*r/ N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. associate-/l/ N/A

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

   6. metadata-eval N/A

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

   7. associate-*r/ N/A

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

   8. div-add N/A

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

   9. lower-/.f64 N/A

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

   10. +-commutative N/A

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

   11. metadata-eval N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower--.f64 N/A

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

   16. associate-*r/ N/A

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

   17. metadata-eval N/A

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

   18. lower-/.f64 99.2

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

4. Applied rewrites 99.2%

   \[\leadsto \frac{x}{y} + \color{blue}{\left(\frac{\frac{2}{z} - -2}{t} - 2\right)} \]
5. Add Preprocessing

Alternative 2: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+16}:\\ \;\;\;\;\frac{2}{t} - \left(2 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{y} + \left(\frac{\frac{2}{z}}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4e+16)
   (- (/ 2.0 t) (- 2.0 (/ x y)))
   (if (<= z 1.8e-18)
     (+ (/ x y) (- (/ (/ 2.0 z) t) 2.0))
     (fma (/ (- 1.0 t) t) 2.0 (/ x y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.4e+16) {
		tmp = (2.0 / t) - (2.0 - (x / y));
	} else if (z <= 1.8e-18) {
		tmp = (x / y) + (((2.0 / z) / t) - 2.0);
	} else {
		tmp = fma(((1.0 - t) / t), 2.0, (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.4e+16)
		tmp = Float64(Float64(2.0 / t) - Float64(2.0 - Float64(x / y)));
	elseif (z <= 1.8e-18)
		tmp = Float64(Float64(x / y) + Float64(Float64(Float64(2.0 / z) / t) - 2.0));
	else
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.4e+16], N[(N[(2.0 / t), $MachinePrecision] - N[(2.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.8e-18], N[(N[(x / y), $MachinePrecision] + N[(N[(N[(2.0 / z), $MachinePrecision] / t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4e16

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -5.4e16 < z < 1.80000000000000005e-18

   1. Initial program 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. associate-/l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. div-add N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. metadata-eval N/A

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

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

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lower--.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 99.2

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

   4. Applied rewrites 99.2%

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

   5. Taylor expanded in z around 0

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

      1. lift-/.f64 81.1

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

   7. Applied rewrites 81.1%

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

   if 1.80000000000000005e-18 < z

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

Alternative 3: 92.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ \mathbf{if}\;z \leq -75:\\ \;\;\;\;\frac{2}{t} - \left(2 - \frac{x}{y}\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{y} + t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z))))
   (if (<= z -75.0)
     (- (/ 2.0 t) (- 2.0 (/ x y)))
     (if (<= z 3.8e-110)
       (+ (/ x y) t_1)
       (if (<= z 8e-22)
         (fma -1.0 2.0 t_1)
         (fma (/ (- 1.0 t) t) 2.0 (/ x y)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double tmp;
	if (z <= -75.0) {
		tmp = (2.0 / t) - (2.0 - (x / y));
	} else if (z <= 3.8e-110) {
		tmp = (x / y) + t_1;
	} else if (z <= 8e-22) {
		tmp = fma(-1.0, 2.0, t_1);
	} else {
		tmp = fma(((1.0 - t) / t), 2.0, (x / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	tmp = 0.0
	if (z <= -75.0)
		tmp = Float64(Float64(2.0 / t) - Float64(2.0 - Float64(x / y)));
	elseif (z <= 3.8e-110)
		tmp = Float64(Float64(x / y) + t_1);
	elseif (z <= 8e-22)
		tmp = fma(-1.0, 2.0, t_1);
	else
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, Float64(x / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -75.0], N[(N[(2.0 / t), $MachinePrecision] - N[(2.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-110], N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[z, 8e-22], N[(-1.0 * 2.0 + t$95$1), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + N[(x / y), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -75

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -75 < z < 3.7999999999999998e-110

   1. Initial program 86.9%

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

   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 62.6%

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

   if 3.7999999999999998e-110 < z < 8.0000000000000004e-22

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 48.8%

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

   if 8.0000000000000004e-22 < z

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 71.3

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

   4. Applied rewrites 71.3%

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

Alternative 4: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{2}{t} - \left(2 - \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -75:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{y} + t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z))) (t_2 (- (/ 2.0 t) (- 2.0 (/ x y)))))
   (if (<= z -75.0)
     t_2
     (if (<= z 3.8e-110)
       (+ (/ x y) t_1)
       (if (<= z 8e-22) (fma -1.0 2.0 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 = (2.0 / t) - (2.0 - (x / y));
	double tmp;
	if (z <= -75.0) {
		tmp = t_2;
	} else if (z <= 3.8e-110) {
		tmp = (x / y) + t_1;
	} else if (z <= 8e-22) {
		tmp = fma(-1.0, 2.0, 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(2.0 / t) - Float64(2.0 - Float64(x / y)))
	tmp = 0.0
	if (z <= -75.0)
		tmp = t_2;
	elseif (z <= 3.8e-110)
		tmp = Float64(Float64(x / y) + t_1);
	elseif (z <= 8e-22)
		tmp = fma(-1.0, 2.0, t_1);
	else
		tmp = t_2;
	end
	return 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[(2.0 / t), $MachinePrecision] - N[(2.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -75.0], t$95$2, If[LessEqual[z, 3.8e-110], N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[z, 8e-22], N[(-1.0 * 2.0 + t$95$1), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -75 or 8.0000000000000004e-22 < z

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -75 < z < 3.7999999999999998e-110

   1. Initial program 86.9%

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

   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 62.6%

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

   if 3.7999999999999998e-110 < z < 8.0000000000000004e-22

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 48.8%

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

Alternative 5: 91.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 0.004:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, t, \frac{2}{z}\right) + 2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 (* t z)))))
   (if (<= (/ x y) -2e+25)
     t_1
     (if (<= (/ x y) 0.004) (/ (+ (fma -2.0 t (/ 2.0 z)) 2.0) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / (t * z));
	double tmp;
	if ((x / y) <= -2e+25) {
		tmp = t_1;
	} else if ((x / y) <= 0.004) {
		tmp = (fma(-2.0, t, (2.0 / z)) + 2.0) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)))
	tmp = 0.0
	if (Float64(x / y) <= -2e+25)
		tmp = t_1;
	elseif (Float64(x / y) <= 0.004)
		tmp = Float64(Float64(fma(-2.0, t, Float64(2.0 / z)) + 2.0) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -2e+25], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 0.004], N[(N[(N[(-2.0 * t + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2.00000000000000018e25 or 0.0040000000000000001 < (/.f64 x y)

   1. Initial program 86.9%

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

   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 62.6%

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

   if -2.00000000000000018e25 < (/.f64 x y) < 0.0040000000000000001

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-+.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 66.6

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

   7. Applied rewrites 66.6%

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

Alternative 6: 85.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t} - \left(2 - \frac{x}{y}\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, \frac{2}{t \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ 2.0 t) (- 2.0 (/ x y)))))
   (if (<= z -2.4e-16)
     t_1
     (if (<= z 8e-22) (fma -1.0 2.0 (/ 2.0 (* t z))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 / t) - (2.0 - (x / y));
	double tmp;
	if (z <= -2.4e-16) {
		tmp = t_1;
	} else if (z <= 8e-22) {
		tmp = fma(-1.0, 2.0, (2.0 / (t * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 / t) - Float64(2.0 - Float64(x / y)))
	tmp = 0.0
	if (z <= -2.4e-16)
		tmp = t_1;
	elseif (z <= 8e-22)
		tmp = fma(-1.0, 2.0, 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[(N[(2.0 / t), $MachinePrecision] - N[(2.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.4e-16], t$95$1, If[LessEqual[z, 8e-22], N[(-1.0 * 2.0 + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.40000000000000005e-16 or 8.0000000000000004e-22 < z

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. lift-/.f64 N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. times-frac N/A

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

      9. mul-1-neg N/A

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

      10. mul-1-neg N/A

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

      11. frac-2neg N/A

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

      12. lower--.f64 N/A

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

      13. lift-/.f64 71.3

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

   9. Applied rewrites 71.3%

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

   if -2.40000000000000005e-16 < z < 8.0000000000000004e-22

   1. Initial program 86.9%

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

   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. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. lift-*.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 48.8%

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

Alternative 7: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} - 2\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 500000000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

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 -2e+75)
     t_1
     (if (<= t_2 500000000.0) 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 <= -2e+75) {
		tmp = t_1;
	} else if (t_2 <= 500000000.0) {
		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 <= -2e+75) {
		tmp = t_1;
	} else if (t_2 <= 500000000.0) {
		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 <= -2e+75:
		tmp = t_1
	elif t_2 <= 500000000.0:
		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 <= -2e+75)
		tmp = t_1;
	elseif (t_2 <= 500000000.0)
		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 <= -2e+75)
		tmp = t_1;
	elseif (t_2 <= 500000000.0)
		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, -2e+75], t$95$1, If[LessEqual[t$95$2, 500000000.0], 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)) < -1.99999999999999985e75 or 5e8 < (/.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 86.9%

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

   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. +-commutative N/A

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

      3. metadata-eval N/A

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

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

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. lower--.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 48.3

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

   4. Applied rewrites 48.3%

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

   if -1.99999999999999985e75 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e8 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 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 8: 68.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ t_3 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -4 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 10^{+182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 -4e+80)
     t_1
     (if (<= t_3 1e+182) 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 <= -4e+80) {
		tmp = t_1;
	} else if (t_3 <= 1e+182) {
		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 <= -4e+80) {
		tmp = t_1;
	} else if (t_3 <= 1e+182) {
		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 <= -4e+80:
		tmp = t_1
	elif t_3 <= 1e+182:
		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 <= -4e+80)
		tmp = t_1;
	elseif (t_3 <= 1e+182)
		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 <= -4e+80)
		tmp = t_1;
	elseif (t_3 <= 1e+182)
		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, -4e+80], t$95$1, If[LessEqual[t$95$3, 1e+182], 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)) < -4e80 or 1.0000000000000001e182 < (/.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 86.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 30.8

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

   4. Applied rewrites 30.8%

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

   if -4e80 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 1.0000000000000001e182 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 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 9: 65.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \cdot 10^{+25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{t} - 2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -2e+25)
   (/ x y)
   (if (<= (/ x y) 2e-20) (- (/ 2.0 t) 2.0) (- (/ x y) 2.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e+25) {
		tmp = x / y;
	} else if ((x / y) <= 2e-20) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = (x / y) - 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 ((x / y) <= (-2d+25)) then
        tmp = x / y
    else if ((x / y) <= 2d-20) then
        tmp = (2.0d0 / t) - 2.0d0
    else
        tmp = (x / y) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -2e+25) {
		tmp = x / y;
	} else if ((x / y) <= 2e-20) {
		tmp = (2.0 / t) - 2.0;
	} else {
		tmp = (x / y) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -2e+25:
		tmp = x / y
	elif (x / y) <= 2e-20:
		tmp = (2.0 / t) - 2.0
	else:
		tmp = (x / y) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -2e+25)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 2e-20)
		tmp = Float64(Float64(2.0 / t) - 2.0);
	else
		tmp = Float64(Float64(x / y) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -2e+25)
		tmp = x / y;
	elseif ((x / y) <= 2e-20)
		tmp = (2.0 / t) - 2.0;
	else
		tmp = (x / y) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -2e+25], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e-20], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -2.00000000000000018e25

   1. Initial program 86.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 35.5

         \[\leadsto \frac{x}{\color{blue}{y}} \]

   7. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -2.00000000000000018e25 < (/.f64 x y) < 1.99999999999999989e-20

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 37.7

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

   9. Applied rewrites 37.7%

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

   10. Taylor expanded in t around inf

       \[\leadsto 2 \cdot \frac{1}{t} - 2 \]
   11. Step-by-step derivation

       1. lower--.f64 N/A

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

       2. associate-*r/ N/A

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

       3. metadata-eval N/A

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

       4. lower-/.f64 37.7

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

   12. Applied rewrites 37.7%

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

   if 1.99999999999999989e-20 < (/.f64 x y)

   1. Initial program 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

Alternative 10: 60.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-119}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ x y) 2.0)))
   (if (<= t -1.45e-155) t_1 (if (<= t 2.1e-119) (/ 2.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double tmp;
	if (t <= -1.45e-155) {
		tmp = t_1;
	} else if (t <= 2.1e-119) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x / y) - 2.0d0
    if (t <= (-1.45d-155)) then
        tmp = t_1
    else if (t <= 2.1d-119) then
        tmp = 2.0d0 / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double tmp;
	if (t <= -1.45e-155) {
		tmp = t_1;
	} else if (t <= 2.1e-119) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) - 2.0
	tmp = 0
	if t <= -1.45e-155:
		tmp = t_1
	elif t <= 2.1e-119:
		tmp = 2.0 / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) - 2.0)
	tmp = 0.0
	if (t <= -1.45e-155)
		tmp = t_1;
	elseif (t <= 2.1e-119)
		tmp = Float64(2.0 / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) - 2.0;
	tmp = 0.0;
	if (t <= -1.45e-155)
		tmp = t_1;
	elseif (t <= 2.1e-119)
		tmp = 2.0 / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[t, -1.45e-155], t$95$1, If[LessEqual[t, 2.1e-119], N[(2.0 / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.45000000000000005e-155 or 2.1e-119 < t

   1. Initial program 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

   if -1.45000000000000005e-155 < t < 2.1e-119

   1. Initial program 86.9%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   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. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. frac-add N/A

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

      10. associate-/r* N/A

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

      11. lower-/.f64 N/A

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

   3. Applied rewrites 68.0%

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

   4. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. lower-*.f64 54.1

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

   6. Applied rewrites 54.1%

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

   7. Taylor expanded in t around 0

      \[\leadsto -\frac{-2}{t} \]
   8. Step-by-step derivation

      1. lower-/.f64 19.7

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

   9. Applied rewrites 19.7%

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

   10. Taylor expanded in t around 0

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

       1. lower-/.f64 19.7

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

   12. Applied rewrites 19.7%

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

Alternative 11: 52.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -10:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.0001:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -10.0) (/ x y) (if (<= (/ x y) 0.0001) -2.0 (/ x y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -10.0) {
		tmp = x / y;
	} else if ((x / y) <= 0.0001) {
		tmp = -2.0;
	} else {
		tmp = x / 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) :: tmp
    if ((x / y) <= (-10.0d0)) then
        tmp = x / y
    else if ((x / y) <= 0.0001d0) then
        tmp = -2.0d0
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -10.0) {
		tmp = x / y;
	} else if ((x / y) <= 0.0001) {
		tmp = -2.0;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x / y) <= -10.0:
		tmp = x / y
	elif (x / y) <= 0.0001:
		tmp = -2.0
	else:
		tmp = x / y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -10.0)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 0.0001)
		tmp = -2.0;
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x / y) <= -10.0)
		tmp = x / y;
	elseif ((x / y) <= 0.0001)
		tmp = -2.0;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -10.0], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 0.0001], -2.0, N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -10 or 1.00000000000000005e-4 < (/.f64 x y)

   1. Initial program 86.9%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 80.9%

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

   5. Taylor expanded in x around inf

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

      1. lower-/.f64 35.5

         \[\leadsto \frac{x}{\color{blue}{y}} \]

   7. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{x}{y}} \]

   if -10 < (/.f64 x y) < 1.00000000000000005e-4

   1. Initial program 86.9%

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

   2. Taylor expanded in t around inf

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

      1. lower--.f64 N/A

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

      2. lift-/.f64 53.7

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 20.2%

      \[\leadsto -2 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 20.2% accurate, 24.7× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

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 86.9%

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

2. Taylor expanded in t around inf

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

   1. lower--.f64 N/A

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

   2. lift-/.f64 53.7

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

4. Applied rewrites 53.7%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 20.2%

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

Reproduce

herbie shell --seed 2025128 
(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))))