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

Percentage Accurate: 86.3% → 99.1%

Time: 6.4s

Alternatives: 18

Speedup: 1.1×

• 23.7% 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: 86.3% 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: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.5%

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

3. Taylor expanded in t around inf

   \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
4. 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. metadata-eval N/A

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

   3. associate-*r/ N/A

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

   4. lower-+.f64 N/A

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

   5. associate-*r/ N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. lower-/.f64 N/A

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

   11. lift-*.f64 99.5

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

5. Applied rewrites 99.5%

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

Alternative 2: 85.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

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))))
   (if (<= t_2 -2e+115)
     t_1
     (if (<= t_2 5e+16)
       (+ (/ x y) (- (/ 2.0 t) 2.0))
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

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 tmp;
	if (t_2 <= -2e+115) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = (x / y) + ((2.0 / t) - 2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	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 tmp;
	if (t_2 <= -2e+115) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = (x / y) + ((2.0 / t) - 2.0);
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	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)
	tmp = 0
	if t_2 <= -2e+115:
		tmp = t_1
	elif t_2 <= 5e+16:
		tmp = (x / y) + ((2.0 / t) - 2.0)
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = (x / y) + -2.0
	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))
	tmp = 0.0
	if (t_2 <= -2e+115)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = Float64(Float64(x / y) + Float64(Float64(2.0 / t) - 2.0));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	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);
	tmp = 0.0;
	if (t_2 <= -2e+115)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = (x / y) + ((2.0 / t) - 2.0);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = (x / y) + -2.0;
	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]}, If[LessEqual[t$95$2, -2e+115], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2e115 or 5e16 < (/.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 98.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   4. 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 76.7

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

   5. Applied rewrites 76.7%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 94.1

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

   8. Applied rewrites 94.1%

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

   if +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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 3: 82.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

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))))
   (if (<= t_2 -2e+115)
     t_1
     (if (<= t_2 5e+16)
       (/ (fma -2.0 y x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

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 tmp;
	if (t_2 <= -2e+115) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = fma(-2.0, y, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	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))
	tmp = 0.0
	if (t_2 <= -2e+115)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = Float64(fma(-2.0, y, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+115], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2e115 or 5e16 < (/.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 98.3%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   4. 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 76.7

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

   5. Applied rewrites 76.7%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 85.1

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

   11. Applied rewrites 85.1%

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

   if +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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 4: 82.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{fma}\left(z, 2, 2\right)}{t \cdot z}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (fma z 2.0 2.0) (* t z)))
        (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_2 -2e+115)
     t_1
     (if (<= t_2 5e+16)
       (/ (fma -2.0 y x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, 2.0, 2.0) / (t * z);
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_2 <= -2e+115) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = fma(-2.0, y, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(fma(z, 2.0, 2.0) / Float64(t * z))
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_2 <= -2e+115)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = Float64(fma(-2.0, y, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * 2.0 + 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $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]}, If[LessEqual[t$95$2, -2e+115], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2e115 or 5e16 < (/.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 98.3%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Add Preprocessing
   3. 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

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

      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

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

      9. +-commutative N/A

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

      10. associate-/r* N/A

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

      11. frac-add N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 75.3%

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

   5. Taylor expanded in t around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. distribute-lft-in N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 85.1

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

   11. Applied rewrites 85.1%

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

   if +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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 5: 68.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+115}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_1 -2e+115)
     (/ 2.0 (* t z))
     (if (<= t_1 5e+16)
       (/ (fma -2.0 y x) y)
       (if (<= t_1 INFINITY) (/ (/ 2.0 t) z) (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_1 <= -2e+115) {
		tmp = 2.0 / (t * z);
	} else if (t_1 <= 5e+16) {
		tmp = fma(-2.0, y, x) / y;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (2.0 / t) / z;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_1 <= -2e+115)
		tmp = Float64(2.0 / Float64(t * z));
	elseif (t_1 <= 5e+16)
		tmp = Float64(fma(-2.0, y, x) / y);
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(2.0 / t) / z);
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+115], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e+16], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(2.0 / t), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2e115

   1. Initial program 96.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 54.3

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

   5. Applied rewrites 54.3%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 85.1

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

   11. Applied rewrites 85.1%

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

   if 5e16 < (/.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 99.7%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 53.2

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

   5. Applied rewrites 53.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 53.2

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

   7. Applied rewrites 53.2%

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

   if +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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 6: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+115} \lor \neg \left(t\_1 \leq 5 \cdot 10^{+16} \lor \neg \left(t\_1 \leq \infty\right)\right):\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (or (<= t_1 -2e+115) (not (or (<= t_1 5e+16) (not (<= t_1 INFINITY)))))
     (/ 2.0 (* t z))
     (+ (/ x y) -2.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+115) || !((t_1 <= 5e+16) || !(t_1 <= ((double) INFINITY)))) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if ((t_1 <= -2e+115) || !((t_1 <= 5e+16) || !(t_1 <= Double.POSITIVE_INFINITY))) {
		tmp = 2.0 / (t * z);
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if (t_1 <= -2e+115) or not ((t_1 <= 5e+16) or not (t_1 <= math.inf)):
		tmp = 2.0 / (t * z)
	else:
		tmp = (x / y) + -2.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if ((t_1 <= -2e+115) || !((t_1 <= 5e+16) || !(t_1 <= Inf)))
		tmp = Float64(2.0 / Float64(t * z));
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if ((t_1 <= -2e+115) || ~(((t_1 <= 5e+16) || ~((t_1 <= Inf)))))
		tmp = 2.0 / (t * z);
	else
		tmp = (x / y) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -2e+115], N[Not[Or[LessEqual[t$95$1, 5e+16], N[Not[LessEqual[t$95$1, Infinity]], $MachinePrecision]]], $MachinePrecision]], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]

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)) < -2e115 or 5e16 < (/.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 98.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16 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 77.2%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 88.4%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq -2 \cdot 10^{+115} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq 5 \cdot 10^{+16} \lor \neg \left(\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \leq \infty\right)\right):\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+115}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z)))
        (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_2 -2e+115)
     t_1
     (if (<= t_2 5e+16)
       (/ (fma -2.0 y x) y)
       (if (<= t_2 INFINITY) t_1 (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_2 <= -2e+115) {
		tmp = t_1;
	} else if (t_2 <= 5e+16) {
		tmp = fma(-2.0, y, x) / y;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_2 <= -2e+115)
		tmp = t_1;
	elseif (t_2 <= 5e+16)
		tmp = Float64(fma(-2.0, y, x) / y);
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / y) + -2.0);
	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 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+115], t$95$1, If[LessEqual[t$95$2, 5e+16], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -2e115 or 5e16 < (/.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 98.3%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lift-*.f64 53.7

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

   5. Applied rewrites 53.7%

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

   if -2e115 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e16

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 85.1

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

   11. Applied rewrites 85.1%

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

   if +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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 8: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (/.f64 x y) (/.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 99.4%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. distribute-rgt-out N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   if +inf.0 < (+.f64 (/.f64 x y) (/.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 0.0%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 100.0%

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

Alternative 9: 93.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1000000000 \lor \neg \left(\frac{x}{y} \leq 10\right):\\ \;\;\;\;\frac{x}{y} + \frac{\frac{\mathsf{fma}\left(z, 2, 2\right)}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{2}{t \cdot z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -1000000000.0) (not (<= (/ x y) 10.0)))
   (+ (/ x y) (/ (/ (fma z 2.0 2.0) t) z))
   (fma (/ (- 1.0 t) t) 2.0 (/ 2.0 (* t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -1000000000.0) || !((x / y) <= 10.0)) {
		tmp = (x / y) + ((fma(z, 2.0, 2.0) / t) / z);
	} else {
		tmp = fma(((1.0 - t) / t), 2.0, (2.0 / (t * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -1000000000.0) || !(Float64(x / y) <= 10.0))
		tmp = Float64(Float64(x / y) + Float64(Float64(fma(z, 2.0, 2.0) / t) / z));
	else
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, Float64(2.0 / Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -1000000000.0], N[Not[LessEqual[N[(x / y), $MachinePrecision], 10.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + N[(N[(N[(z * 2.0 + 2.0), $MachinePrecision] / t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1e9 or 10 < (/.f64 x y)

   1. Initial program 90.3%

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

   3. Taylor expanded in t around 0

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

      1. associate-/r* N/A

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

      2. div-add N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate-*r/ N/A

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

      10. metadata-eval N/A

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

      11. associate-*r/ N/A

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

      12. div-add N/A

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

      13. lower-/.f64 N/A

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

      14. +-commutative N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -1e9 < (/.f64 x y) < 10

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 94.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1000000000 \lor \neg \left(\frac{x}{y} \leq 10\right):\\ \;\;\;\;\frac{x}{y} + \frac{\frac{\mathsf{fma}\left(z, 2, 2\right)}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{2}{t \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -1000000000 \lor \neg \left(\frac{x}{y} \leq 10\right):\\ \;\;\;\;\frac{x}{y} + t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z))))
   (if (or (<= (/ x y) -1000000000.0) (not (<= (/ x y) 10.0)))
     (+ (/ x y) t_1)
     (fma (/ (- 1.0 t) t) 2.0 t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double tmp;
	if (((x / y) <= -1000000000.0) || !((x / y) <= 10.0)) {
		tmp = (x / y) + t_1;
	} else {
		tmp = fma(((1.0 - t) / t), 2.0, t_1);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	tmp = 0.0
	if ((Float64(x / y) <= -1000000000.0) || !(Float64(x / y) <= 10.0))
		tmp = Float64(Float64(x / y) + t_1);
	else
		tmp = fma(Float64(Float64(1.0 - t) / t), 2.0, t_1);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[N[(x / y), $MachinePrecision], -1000000000.0], N[Not[LessEqual[N[(x / y), $MachinePrecision], 10.0]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(1.0 - t), $MachinePrecision] / t), $MachinePrecision] * 2.0 + t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -1e9 or 10 < (/.f64 x y)

   1. Initial program 90.3%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 90.9%

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

   if -1e9 < (/.f64 x y) < 10

   1. Initial program 86.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 97.2

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

   5. Applied rewrites 97.2%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1000000000 \lor \neg \left(\frac{x}{y} \leq 10\right):\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1 - t}{t}, 2, \frac{2}{t \cdot z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - t, z, 1\right) \cdot 2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e+139)
   (/ x y)
   (if (<= (/ x y) 5e+71)
     (/ (* (fma (- 1.0 t) z 1.0) 2.0) (* t z))
     (+ (/ x y) (/ 2.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e+139) {
		tmp = x / y;
	} else if ((x / y) <= 5e+71) {
		tmp = (fma((1.0 - t), z, 1.0) * 2.0) / (t * z);
	} else {
		tmp = (x / y) + (2.0 / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1e+139)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 5e+71)
		tmp = Float64(Float64(fma(Float64(1.0 - t), z, 1.0) * 2.0) / Float64(t * z));
	else
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -1e+139], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e+71], N[(N[(N[(N[(1.0 - t), $MachinePrecision] * z + 1.0), $MachinePrecision] * 2.0), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.00000000000000003e139

   1. Initial program 88.8%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000003e139 < (/.f64 x y) < 4.99999999999999972e71

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-*r/ N/A

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

      3. metadata-eval N/A

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

      4. associate-*r/ N/A

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

      5. div-add-rev N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-/r* N/A

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

      9. lower-/.f64 N/A

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

   8. Applied rewrites 79.8%

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

   if 4.99999999999999972e71 < (/.f64 x y)

   1. Initial program 89.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 88.8

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

   8. Applied rewrites 88.8%

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

   9. Taylor expanded in t around 0

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

       1. lift-/.f64 88.8

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

   11. Applied rewrites 88.8%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\frac{\mathsf{fma}\left(1 - t, z, 1\right) \cdot 2}{t \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+139}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, \frac{2}{t \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -1e+139)
   (/ x y)
   (if (<= (/ x y) 5e+71)
     (fma -1.0 2.0 (/ 2.0 (* t z)))
     (/ (fma -2.0 y x) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -1e+139) {
		tmp = x / y;
	} else if ((x / y) <= 5e+71) {
		tmp = fma(-1.0, 2.0, (2.0 / (t * z)));
	} else {
		tmp = fma(-2.0, y, x) / y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -1e+139)
		tmp = Float64(x / y);
	elseif (Float64(x / y) <= 5e+71)
		tmp = fma(-1.0, 2.0, Float64(2.0 / Float64(t * z)));
	else
		tmp = Float64(fma(-2.0, y, x) / y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -1e+139], N[(x / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 5e+71], N[(-1.0 * 2.0 + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -1.00000000000000003e139

   1. Initial program 88.8%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1.00000000000000003e139 < (/.f64 x y) < 4.99999999999999972e71

   1. Initial program 88.0%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 67.2%

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

   if 4.99999999999999972e71 < (/.f64 x y)

   1. Initial program 89.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 98.2

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

   5. Applied rewrites 98.2%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 76.7

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

   11. Applied rewrites 76.7%

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

Alternative 13: 65.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.9 \cdot 10^{-7} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -4.9e-7) (not (<= (/ x y) 6.6e-6)))
   (+ (/ x y) -2.0)
   (- (/ 2.0 t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -4.9e-7) || !((x / y) <= 6.6e-6)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = (2.0 / t) - 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) <= (-4.9d-7)) .or. (.not. ((x / y) <= 6.6d-6))) then
        tmp = (x / y) + (-2.0d0)
    else
        tmp = (2.0d0 / t) - 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) <= -4.9e-7) || !((x / y) <= 6.6e-6)) {
		tmp = (x / y) + -2.0;
	} else {
		tmp = (2.0 / t) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -4.9e-7) or not ((x / y) <= 6.6e-6):
		tmp = (x / y) + -2.0
	else:
		tmp = (2.0 / t) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -4.9e-7) || !(Float64(x / y) <= 6.6e-6))
		tmp = Float64(Float64(x / y) + -2.0);
	else
		tmp = Float64(Float64(2.0 / t) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -4.9e-7) || ~(((x / y) <= 6.6e-6)))
		tmp = (x / y) + -2.0;
	else
		tmp = (2.0 / t) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -4.9e-7], N[Not[LessEqual[N[(x / y), $MachinePrecision], 6.6e-6]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -4.8999999999999997e-7 or 6.60000000000000034e-6 < (/.f64 x y)

   1. Initial program 89.4%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 70.9%

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

   if -4.8999999999999997e-7 < (/.f64 x y) < 6.60000000000000034e-6

   1. Initial program 87.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 61.4

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

   8. Applied rewrites 61.4%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -4.9 \cdot 10^{-7} \lor \neg \left(\frac{x}{y} \leq 6.6 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{x}{y} + -2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6400 \lor \neg \left(\frac{x}{y} \leq 560000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -6400.0) (not (<= (/ x y) 560000.0)))
   (/ x y)
   (- (/ 2.0 t) 2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -6400.0) || !((x / y) <= 560000.0)) {
		tmp = x / y;
	} else {
		tmp = (2.0 / t) - 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) <= (-6400.0d0)) .or. (.not. ((x / y) <= 560000.0d0))) then
        tmp = x / y
    else
        tmp = (2.0d0 / t) - 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) <= -6400.0) || !((x / y) <= 560000.0)) {
		tmp = x / y;
	} else {
		tmp = (2.0 / t) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -6400.0) or not ((x / y) <= 560000.0):
		tmp = x / y
	else:
		tmp = (2.0 / t) - 2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -6400.0) || !(Float64(x / y) <= 560000.0))
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(2.0 / t) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -6400.0) || ~(((x / y) <= 560000.0)))
		tmp = x / y;
	else
		tmp = (2.0 / t) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -6400.0], N[Not[LessEqual[N[(x / y), $MachinePrecision], 560000.0]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -6400 or 5.6e5 < (/.f64 x y)

   1. Initial program 90.4%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -6400 < (/.f64 x y) < 5.6e5

   1. Initial program 86.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 97.2

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

   5. Applied rewrites 97.2%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 60.1

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

   8. Applied rewrites 60.1%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -6400 \lor \neg \left(\frac{x}{y} \leq 560000\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-2, y, x\right)}{y}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{2}{z} - -2}{t}\\ \mathbf{elif}\;t \leq 75000:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t}\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-1, 2, \frac{2}{t \cdot z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.7e+43)
   (/ (fma -2.0 y x) y)
   (if (<= t 1.4e-171)
     (/ (- (/ 2.0 z) -2.0) t)
     (if (<= t 75000.0)
       (+ (/ x y) (/ 2.0 t))
       (if (<= t 5.8e+64) (fma -1.0 2.0 (/ 2.0 (* t z))) (+ (/ x y) -2.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.7e+43) {
		tmp = fma(-2.0, y, x) / y;
	} else if (t <= 1.4e-171) {
		tmp = ((2.0 / z) - -2.0) / t;
	} else if (t <= 75000.0) {
		tmp = (x / y) + (2.0 / t);
	} else if (t <= 5.8e+64) {
		tmp = fma(-1.0, 2.0, (2.0 / (t * z)));
	} else {
		tmp = (x / y) + -2.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.7e+43)
		tmp = Float64(fma(-2.0, y, x) / y);
	elseif (t <= 1.4e-171)
		tmp = Float64(Float64(Float64(2.0 / z) - -2.0) / t);
	elseif (t <= 75000.0)
		tmp = Float64(Float64(x / y) + Float64(2.0 / t));
	elseif (t <= 5.8e+64)
		tmp = fma(-1.0, 2.0, Float64(2.0 / Float64(t * z)));
	else
		tmp = Float64(Float64(x / y) + -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.7e+43], N[(N[(-2.0 * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 1.4e-171], N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 75000.0], N[(N[(x / y), $MachinePrecision] + N[(2.0 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.8e+64], N[(-1.0 * 2.0 + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + -2.0), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if t < -2.7000000000000002e43

   1. Initial program 71.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 96.8%

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

   9. Taylor expanded in t around inf

      \[\leadsto \frac{x + -2 \cdot y}{y} \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. lower-fma.f64 92.0

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

   11. Applied rewrites 92.0%

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

   if -2.7000000000000002e43 < t < 1.40000000000000011e-171

   1. Initial program 98.8%

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

   3. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   4. 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 79.0

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

   5. Applied rewrites 79.0%

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

   if 1.40000000000000011e-171 < t < 75000

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 83.2

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

   8. Applied rewrites 83.2%

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

   9. Taylor expanded in t around 0

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

       1. lift-/.f64 80.5

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

   11. Applied rewrites 80.5%

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

   if 75000 < t < 5.79999999999999986e64

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.1%

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

   if 5.79999999999999986e64 < t

   1. Initial program 76.1%

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

   3. Taylor expanded in t around inf

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

   5. Applied rewrites 91.8%

      \[\leadsto \frac{x}{y} + \color{blue}{-2} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \lor \neg \left(\frac{x}{y} \leq 0.0014\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ x y) -2.0) (not (<= (/ x y) 0.0014))) (/ x y) -2.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -2.0) || !((x / y) <= 0.0014)) {
		tmp = x / y;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((x / y) <= (-2.0d0)) .or. (.not. ((x / y) <= 0.0014d0))) then
        tmp = x / y
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((x / y) <= -2.0) || !((x / y) <= 0.0014)) {
		tmp = x / y;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((x / y) <= -2.0) or not ((x / y) <= 0.0014):
		tmp = x / y
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(x / y) <= -2.0) || !(Float64(x / y) <= 0.0014))
		tmp = Float64(x / y);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((x / y) <= -2.0) || ~(((x / y) <= 0.0014)))
		tmp = x / y;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(x / y), $MachinePrecision], -2.0], N[Not[LessEqual[N[(x / y), $MachinePrecision], 0.0014]], $MachinePrecision]], N[(x / y), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -2 or 0.00139999999999999999 < (/.f64 x y)

   1. Initial program 90.6%

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

   3. Taylor expanded in x around inf

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

      1. lift-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -2 < (/.f64 x y) < 0.00139999999999999999

   1. Initial program 86.1%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   4. 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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 36.4%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -2 \lor \neg \left(\frac{x}{y} \leq 0.0014\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 92.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-6} \lor \neg \left(z \leq 0.07\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.2e-6) (not (<= z 0.07)))
   (+ (/ x y) (- (/ 2.0 t) 2.0))
   (+ (/ x y) (/ 2.0 (* t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.2e-6) || !(z <= 0.07)) {
		tmp = (x / y) + ((2.0 / t) - 2.0);
	} else {
		tmp = (x / y) + (2.0 / (t * z));
	}
	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 ((z <= (-9.2d-6)) .or. (.not. (z <= 0.07d0))) then
        tmp = (x / y) + ((2.0d0 / t) - 2.0d0)
    else
        tmp = (x / y) + (2.0d0 / (t * z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.2e-6) || !(z <= 0.07)) {
		tmp = (x / y) + ((2.0 / t) - 2.0);
	} else {
		tmp = (x / y) + (2.0 / (t * z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.2e-6) or not (z <= 0.07):
		tmp = (x / y) + ((2.0 / t) - 2.0)
	else:
		tmp = (x / y) + (2.0 / (t * z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.2e-6) || !(z <= 0.07))
		tmp = Float64(Float64(x / y) + Float64(Float64(2.0 / t) - 2.0));
	else
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -9.2e-6) || ~((z <= 0.07)))
		tmp = (x / y) + ((2.0 / t) - 2.0);
	else
		tmp = (x / y) + (2.0 / (t * z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.2e-6], N[Not[LessEqual[z, 0.07]], $MachinePrecision]], N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 / t), $MachinePrecision] - 2.0), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.2e-6 or 0.070000000000000007 < z

   1. Initial program 75.2%

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

   3. Taylor expanded in t around inf

      \[\leadsto \frac{x}{y} + \color{blue}{\left(\left(2 \cdot \frac{1}{t} + \frac{2}{t \cdot z}\right) - 2\right)} \]
   4. 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. metadata-eval N/A

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

      3. associate-*r/ N/A

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

      4. lower-+.f64 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. lift-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. *-inverses N/A

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

      3. distribute-lft-out-- N/A

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

      4. metadata-eval N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r/ N/A

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

      7. metadata-eval N/A

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

      8. lift-/.f64 97.3

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

   8. Applied rewrites 97.3%

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

   if -9.2e-6 < z < 0.070000000000000007

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

   5. Applied rewrites 88.3%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-6} \lor \neg \left(z \leq 0.07\right):\\ \;\;\;\;\frac{x}{y} + \left(\frac{2}{t} - 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.8% accurate, 47.0× 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 88.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
4. 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 63.1

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

5. Applied rewrites 63.1%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 17.8%

   \[\leadsto -2 \]

9. Final simplification 17.8%

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

Developer Target 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64

  :alt
  (! :herbie-platform default (- (/ (+ (/ 2 z) 2) t) (- 2 (/ x y))))

  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))