Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.9% → 99.8%

Time: 10.4s

Alternatives: 15

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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, a)
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), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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, a)
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), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) - -1}, a, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. remove-double-neg N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f64 99.8

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

   15. lift-+.f64 N/A

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

   16. add-flip N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval 99.8

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

3. Applied rewrites 99.8%

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

Alternative 2: 96.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{+132}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{\left(t - z\right) - -1}, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 2.7e+132)
   (fma (/ a (- (- t z) -1.0)) (- z y) x)
   (fma (/ (- z y) (- 1.0 z)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 2.7e+132) {
		tmp = fma((a / ((t - z) - -1.0)), (z - y), x);
	} else {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 2.7e+132)
		tmp = fma(Float64(a / Float64(Float64(t - z) - -1.0)), Float64(z - y), x);
	else
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 2.7e+132], N[(N[(a / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 2.7e132

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

   if 2.7e132 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

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

Alternative 3: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;x - \frac{y - z}{\frac{t + 1}{a}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- 1.0 z)) a x)))
   (if (<= z -5e+23)
     t_1
     (if (<= z 3.7e+34) (- x (/ (- y z) (/ (+ t 1.0) a))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (1.0 - z)), a, x);
	double tmp;
	if (z <= -5e+23) {
		tmp = t_1;
	} else if (z <= 3.7e+34) {
		tmp = x - ((y - z) / ((t + 1.0) / a));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x)
	tmp = 0.0
	if (z <= -5e+23)
		tmp = t_1;
	elseif (z <= 3.7e+34)
		tmp = Float64(x - Float64(Float64(y - z) / Float64(Float64(t + 1.0) / a)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -5e+23], t$95$1, If[LessEqual[z, 3.7e+34], N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(t + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.9999999999999999e23 or 3.70000000000000009e34 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

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

   if -4.9999999999999999e23 < z < 3.70000000000000009e34

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

   4. Applied rewrites 73.9%

      \[\leadsto x - \frac{y - z}{\frac{\color{blue}{t} + 1}{a}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+21}:\\ \;\;\;\;x - \left(-a\right) \cdot \left(y \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 0.0044:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - z\right) - -1}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7e+21)
   (- x (* (- a) (* y (/ -1.0 t))))
   (if (<= t 0.0044)
     (fma (/ (- z y) (- 1.0 z)) a x)
     (fma (/ z (- (- t z) -1.0)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7e+21) {
		tmp = x - (-a * (y * (-1.0 / t)));
	} else if (t <= 0.0044) {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	} else {
		tmp = fma((z / ((t - z) - -1.0)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7e+21)
		tmp = Float64(x - Float64(Float64(-a) * Float64(y * Float64(-1.0 / t))));
	elseif (t <= 0.0044)
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	else
		tmp = fma(Float64(z / Float64(Float64(t - z) - -1.0)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7e+21], N[(x - N[((-a) * N[(y * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 0.0044], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e21

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg-rev N/A

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

      12. lower-/.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if -7e21 < t < 0.00440000000000000027

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

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

   if 0.00440000000000000027 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 73.6%

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

Alternative 5: 88.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+21}:\\ \;\;\;\;x - \left(-a\right) \cdot \left(y \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{1 - z}, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(t - z\right) - -1}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7e+21)
   (- x (* (- a) (* y (/ -1.0 t))))
   (if (<= t 2.7e-5)
     (fma (/ a (- 1.0 z)) (- z y) x)
     (fma (/ z (- (- t z) -1.0)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7e+21) {
		tmp = x - (-a * (y * (-1.0 / t)));
	} else if (t <= 2.7e-5) {
		tmp = fma((a / (1.0 - z)), (z - y), x);
	} else {
		tmp = fma((z / ((t - z) - -1.0)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7e+21)
		tmp = Float64(x - Float64(Float64(-a) * Float64(y * Float64(-1.0 / t))));
	elseif (t <= 2.7e-5)
		tmp = fma(Float64(a / Float64(1.0 - z)), Float64(z - y), x);
	else
		tmp = fma(Float64(z / Float64(Float64(t - z) - -1.0)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7e+21], N[(x - N[((-a) * N[(y * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.7e-5], N[(N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision], N[(N[(z / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e21

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg-rev N/A

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

      12. lower-/.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if -7e21 < t < 2.6999999999999999e-5

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{1 - z}}, a, x\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z} \cdot a + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{z - y}{1 - z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z} \cdot a} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z}} \cdot a + x \]

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

   if 2.6999999999999999e-5 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 73.6%

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

Alternative 6: 86.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7 \cdot 10^{+21}:\\ \;\;\;\;x - \left(-a\right) \cdot \left(y \cdot \frac{-1}{t}\right)\\ \mathbf{elif}\;t \leq 1.46 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{1 - z}, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -7e+21)
   (- x (* (- a) (* y (/ -1.0 t))))
   (if (<= t 1.46e+16)
     (fma (/ a (- 1.0 z)) (- z y) x)
     (fma (/ (- z y) t) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -7e+21) {
		tmp = x - (-a * (y * (-1.0 / t)));
	} else if (t <= 1.46e+16) {
		tmp = fma((a / (1.0 - z)), (z - y), x);
	} else {
		tmp = fma(((z - y) / t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -7e+21)
		tmp = Float64(x - Float64(Float64(-a) * Float64(y * Float64(-1.0 / t))));
	elseif (t <= 1.46e+16)
		tmp = fma(Float64(a / Float64(1.0 - z)), Float64(z - y), x);
	else
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -7e+21], N[(x - N[((-a) * N[(y * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.46e+16], N[(N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7e21

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. frac-2neg N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. distribute-lft-neg-in N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. metadata-eval N/A

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

      11. frac-2neg-rev N/A

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

      12. lower-/.f64 56.7

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

   9. Applied rewrites 56.7%

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

   if -7e21 < t < 1.46e16

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

      \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{1 - z}}, a, x\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z} \cdot a + x} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{a \cdot \frac{z - y}{1 - z}} + x \]

      3. *-commutative N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z} \cdot a} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z - y}{1 - z}} \cdot a + x \]

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*l/ N/A

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

      10. *-lft-identity N/A

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

      11. lower-/.f64 77.8

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

   8. Applied rewrites 77.8%

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

   if 1.46e16 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.2

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

   6. Applied rewrites 54.2%

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

Alternative 7: 84.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+35}:\\ \;\;\;\;x - \frac{a}{t - -1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.2e+77)
   (- x a)
   (if (<= z 1.35e+35)
     (- x (* (/ a (- t -1.0)) y))
     (fma (/ z (- 1.0 z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.2e+77) {
		tmp = x - a;
	} else if (z <= 1.35e+35) {
		tmp = x - ((a / (t - -1.0)) * y);
	} else {
		tmp = fma((z / (1.0 - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.2e+77)
		tmp = Float64(x - a);
	elseif (z <= 1.35e+35)
		tmp = Float64(x - Float64(Float64(a / Float64(t - -1.0)) * y));
	else
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.2e+77], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.35e+35], N[(x - N[(N[(a / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1999999999999999e77

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -1.1999999999999999e77 < z < 1.35000000000000001e35

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      9. lower-/.f64 72.6

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      10. lift-+.f64 N/A

          \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      11. +-commutative N/A

          \[\leadsto x - \frac{a}{t + 1} \cdot y \]

      12. add-flip N/A

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

      13. metadata-eval N/A

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

      14. lower--.f64 72.6

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

   6. Applied rewrites 72.6%

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

   if 1.35000000000000001e35 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z}}{1 - z}, a, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.3%

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

Alternative 8: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{a \cdot y}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+23)
   (- x a)
   (if (<= z 4.6e-281)
     (fma (/ (- z y) t) a x)
     (if (<= z 2.4e+82) (- x (/ (* a y) (- 1.0 z))) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+23) {
		tmp = x - a;
	} else if (z <= 4.6e-281) {
		tmp = fma(((z - y) / t), a, x);
	} else if (z <= 2.4e+82) {
		tmp = x - ((a * y) / (1.0 - z));
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+23)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-281)
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	elseif (z <= 2.4e+82)
		tmp = Float64(x - Float64(Float64(a * y) / Float64(1.0 - z)));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+23], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-281], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 2.4e+82], N[(x - N[(N[(a * y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999999e23 or 2.39999999999999998e82 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -4.9999999999999999e23 < z < 4.59999999999999978e-281

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.2

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

   6. Applied rewrites 54.2%

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

   if 4.59999999999999978e-281 < z < 2.39999999999999998e82

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 63.6

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 63.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+82}:\\ \;\;\;\;x - \frac{y}{1 - z} \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+23)
   (- x a)
   (if (<= z 4.6e-281)
     (fma (/ (- z y) t) a x)
     (if (<= z 2.4e+82) (- x (* (/ y (- 1.0 z)) a)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+23) {
		tmp = x - a;
	} else if (z <= 4.6e-281) {
		tmp = fma(((z - y) / t), a, x);
	} else if (z <= 2.4e+82) {
		tmp = x - ((y / (1.0 - z)) * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+23)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-281)
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	elseif (z <= 2.4e+82)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 - z)) * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+23], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-281], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 2.4e+82], N[(x - N[(N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999999e23 or 2.39999999999999998e82 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -4.9999999999999999e23 < z < 4.59999999999999978e-281

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.2

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

   6. Applied rewrites 54.2%

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

   if 4.59999999999999978e-281 < z < 2.39999999999999998e82

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 63.6

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 63.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - \color{blue}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. associate-/l* N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{1 - z}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-/.f64 65.4

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.4%

      \[\leadsto x - \frac{y}{1 - z} \cdot a \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+23}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-281}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+34}:\\ \;\;\;\;x - \frac{a \cdot y}{1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+23)
   (- x a)
   (if (<= z 4.6e-281)
     (fma (/ (- z y) t) a x)
     (if (<= z 3.05e+34) (- x (/ (* a y) 1.0)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+23) {
		tmp = x - a;
	} else if (z <= 4.6e-281) {
		tmp = fma(((z - y) / t), a, x);
	} else if (z <= 3.05e+34) {
		tmp = x - ((a * y) / 1.0);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+23)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-281)
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	elseif (z <= 3.05e+34)
		tmp = Float64(x - Float64(Float64(a * y) / 1.0));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+23], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-281], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 3.05e+34], N[(x - N[(N[(a * y), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.9999999999999999e23 or 3.04999999999999998e34 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -4.9999999999999999e23 < z < 4.59999999999999978e-281

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.2

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

   6. Applied rewrites 54.2%

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

   if 4.59999999999999978e-281 < z < 3.04999999999999998e34

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 63.6

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 63.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto x - \frac{a \cdot y}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.3%

       \[\leadsto x - \frac{a \cdot y}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+38}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-281}:\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+34}:\\ \;\;\;\;x - \frac{a \cdot y}{1}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e+38)
   (- x a)
   (if (<= z 4.6e-281)
     (- x (* (/ y t) a))
     (if (<= z 3.05e+34) (- x (/ (* a y) 1.0)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+38) {
		tmp = x - a;
	} else if (z <= 4.6e-281) {
		tmp = x - ((y / t) * a);
	} else if (z <= 3.05e+34) {
		tmp = x - ((a * y) / 1.0);
	} else {
		tmp = x - a;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d+38)) then
        tmp = x - a
    else if (z <= 4.6d-281) then
        tmp = x - ((y / t) * a)
    else if (z <= 3.05d+34) then
        tmp = x - ((a * y) / 1.0d0)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+38) {
		tmp = x - a;
	} else if (z <= 4.6e-281) {
		tmp = x - ((y / t) * a);
	} else if (z <= 3.05e+34) {
		tmp = x - ((a * y) / 1.0);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e+38:
		tmp = x - a
	elif z <= 4.6e-281:
		tmp = x - ((y / t) * a)
	elif z <= 3.05e+34:
		tmp = x - ((a * y) / 1.0)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e+38)
		tmp = Float64(x - a);
	elseif (z <= 4.6e-281)
		tmp = Float64(x - Float64(Float64(y / t) * a));
	elseif (z <= 3.05e+34)
		tmp = Float64(x - Float64(Float64(a * y) / 1.0));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e+38)
		tmp = x - a;
	elseif (z <= 4.6e-281)
		tmp = x - ((y / t) * a);
	elseif (z <= 3.05e+34)
		tmp = x - ((a * y) / 1.0);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e+38], N[(x - a), $MachinePrecision], If[LessEqual[z, 4.6e-281], N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.05e+34], N[(x - N[(N[(a * y), $MachinePrecision] / 1.0), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.90000000000000007e38 or 3.04999999999999998e34 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -2.90000000000000007e38 < z < 4.59999999999999978e-281

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      3. associate-/l* N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{t}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      6. lower-/.f64 56.7

         \[\leadsto x - \frac{y}{t} \cdot a \]

   9. Applied rewrites 56.7%

      \[\leadsto x - \frac{y}{t} \cdot a \]

   if 4.59999999999999978e-281 < z < 3.04999999999999998e34

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - \color{blue}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 63.6

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 63.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]

   8. Taylor expanded in z around 0

      \[\leadsto x - \frac{a \cdot y}{1} \]
   9. Step-by-step derivation

   10. Applied rewrites 57.3%

       \[\leadsto x - \frac{a \cdot y}{1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 70.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.1 \cdot 10^{+21}:\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{elif}\;t \leq 1.62 \cdot 10^{+136}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.1e+21)
   (- x (* (/ y t) a))
   (if (<= t 1.62e+136) (fma (/ z (- 1.0 z)) a x) (fma (/ (- z y) t) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.1e+21) {
		tmp = x - ((y / t) * a);
	} else if (t <= 1.62e+136) {
		tmp = fma((z / (1.0 - z)), a, x);
	} else {
		tmp = fma(((z - y) / t), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.1e+21)
		tmp = Float64(x - Float64(Float64(y / t) * a));
	elseif (t <= 1.62e+136)
		tmp = fma(Float64(z / Float64(1.0 - z)), a, x);
	else
		tmp = fma(Float64(Float64(z - y) / t), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -4.1e+21], N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.62e+136], N[(N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.1e21

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      3. associate-/l* N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{t}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      6. lower-/.f64 56.7

         \[\leadsto x - \frac{y}{t} \cdot a \]

   9. Applied rewrites 56.7%

      \[\leadsto x - \frac{y}{t} \cdot a \]

   if -4.1e21 < t < 1.6200000000000001e136

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 79.5

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

   6. Applied rewrites 79.5%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z}}{1 - z}, a, x\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 66.3%

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

   if 1.6200000000000001e136 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.8

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.8

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 54.2

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

   6. Applied rewrites 54.2%

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

Alternative 13: 70.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{+38}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 7.6 \cdot 10^{+34}:\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.9e+38) (- x a) (if (<= z 7.6e+34) (- x (* (/ y t) a)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+38) {
		tmp = x - a;
	} else if (z <= 7.6e+34) {
		tmp = x - ((y / t) * a);
	} else {
		tmp = x - a;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.9d+38)) then
        tmp = x - a
    else if (z <= 7.6d+34) then
        tmp = x - ((y / t) * a)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.9e+38) {
		tmp = x - a;
	} else if (z <= 7.6e+34) {
		tmp = x - ((y / t) * a);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.9e+38:
		tmp = x - a
	elif z <= 7.6e+34:
		tmp = x - ((y / t) * a)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.9e+38)
		tmp = Float64(x - a);
	elseif (z <= 7.6e+34)
		tmp = Float64(x - Float64(Float64(y / t) * a));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.9e+38)
		tmp = x - a;
	elseif (z <= 7.6e+34)
		tmp = x - ((y / t) * a);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.9e+38], N[(x - a), $MachinePrecision], If[LessEqual[z, 7.6e+34], N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.90000000000000007e38 or 7.6000000000000003e34 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -2.90000000000000007e38 < z < 7.6000000000000003e34

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      3. associate-/l* N/A

         \[\leadsto x - a \cdot \frac{y}{\color{blue}{t}} \]

      4. *-commutative N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      5. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{t} \cdot a \]

      6. lower-/.f64 56.7

         \[\leadsto x - \frac{y}{t} \cdot a \]

   9. Applied rewrites 56.7%

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

Alternative 14: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-98}:\\ \;\;\;\;x - \frac{a}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.3e+37) (- x a) (if (<= z 5e-98) (- x (* (/ a t) y)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+37) {
		tmp = x - a;
	} else if (z <= 5e-98) {
		tmp = x - ((a / t) * y);
	} else {
		tmp = x - a;
	}
	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, a)
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), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.3d+37)) then
        tmp = x - a
    else if (z <= 5d-98) then
        tmp = x - ((a / t) * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.3e+37) {
		tmp = x - a;
	} else if (z <= 5e-98) {
		tmp = x - ((a / t) * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.3e+37:
		tmp = x - a
	elif z <= 5e-98:
		tmp = x - ((a / t) * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.3e+37)
		tmp = Float64(x - a);
	elseif (z <= 5e-98)
		tmp = Float64(x - Float64(Float64(a / t) * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.3e+37)
		tmp = x - a;
	elseif (z <= 5e-98)
		tmp = x - ((a / t) * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.3e+37], N[(x - a), $MachinePrecision], If[LessEqual[z, 5e-98], N[(x - N[(N[(a / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000002e37 or 5.00000000000000018e-98 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 59.3%

      \[\leadsto x - \color{blue}{a} \]

   if -2.30000000000000002e37 < z < 5.00000000000000018e-98

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.7

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. lower-*.f64 54.6

         \[\leadsto x - \frac{a \cdot y}{t} \]

   7. Applied rewrites 54.6%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{t}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{t} \]

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

         \[\leadsto x - \frac{a}{t} \cdot y \]

      9. lower-/.f64 56.4

         \[\leadsto x - \frac{a}{t} \cdot y \]

   9. Applied rewrites 56.4%

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

Alternative 15: 59.3% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x - a;
}

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, a)
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), intent (in) :: a
    code = x - a
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return Float64(x - a)
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in z around inf

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

4. Applied rewrites 59.3%

   \[\leadsto x - \color{blue}{a} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025143 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))