Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4

Percentage Accurate: 97.8% → 98.3%

Time: 6.2s

Alternatives: 10

Speedup: 0.7×

• 25.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - x) * (z / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x + ((y - x) * (z / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 5e+134) (fma (- y x) (/ z t) x) (* z (/ (- y x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 5e+134) {
		tmp = fma((y - x), (z / t), x);
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 5e+134)
		tmp = fma(Float64(y - x), Float64(z / t), x);
	else
		tmp = Float64(z * Float64(Float64(y - x) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 5e+134], N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 4.99999999999999981e134

   1. Initial program 99.0%

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

      1. +-commutative 99.0%

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

      2. fma-def 99.0%

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

   3. Simplified 99.0%

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

   if 4.99999999999999981e134 < (/.f64 z t)

   1. Initial program 87.4%

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

   2. Taylor expanded in z around -inf 97.9%

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

   3. Taylor expanded in t around 0 97.9%

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

      1. associate-/l* 87.4%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 2: 63.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+135} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) y)))
   (if (<= (/ z t) -1e-70)
     t_1
     (if (<= (/ z t) 2e-17)
       x
       (if (<= (/ z t) 5e+59)
         t_1
         (if (or (<= (/ z t) 5e+135) (not (<= (/ z t) 5e+184)))
           (* (/ z t) (- x))
           (* z (/ y t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = t_1;
	} else if ((z / t) <= 2e-17) {
		tmp = x;
	} else if ((z / t) <= 5e+59) {
		tmp = t_1;
	} else if (((z / t) <= 5e+135) || !((z / t) <= 5e+184)) {
		tmp = (z / t) * -x;
	} else {
		tmp = z * (y / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / t) * y
    if ((z / t) <= (-1d-70)) then
        tmp = t_1
    else if ((z / t) <= 2d-17) then
        tmp = x
    else if ((z / t) <= 5d+59) then
        tmp = t_1
    else if (((z / t) <= 5d+135) .or. (.not. ((z / t) <= 5d+184))) then
        tmp = (z / t) * -x
    else
        tmp = z * (y / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = t_1;
	} else if ((z / t) <= 2e-17) {
		tmp = x;
	} else if ((z / t) <= 5e+59) {
		tmp = t_1;
	} else if (((z / t) <= 5e+135) || !((z / t) <= 5e+184)) {
		tmp = (z / t) * -x;
	} else {
		tmp = z * (y / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * y
	tmp = 0
	if (z / t) <= -1e-70:
		tmp = t_1
	elif (z / t) <= 2e-17:
		tmp = x
	elif (z / t) <= 5e+59:
		tmp = t_1
	elif ((z / t) <= 5e+135) or not ((z / t) <= 5e+184):
		tmp = (z / t) * -x
	else:
		tmp = z * (y / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * y)
	tmp = 0.0
	if (Float64(z / t) <= -1e-70)
		tmp = t_1;
	elseif (Float64(z / t) <= 2e-17)
		tmp = x;
	elseif (Float64(z / t) <= 5e+59)
		tmp = t_1;
	elseif ((Float64(z / t) <= 5e+135) || !(Float64(z / t) <= 5e+184))
		tmp = Float64(Float64(z / t) * Float64(-x));
	else
		tmp = Float64(z * Float64(y / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * y;
	tmp = 0.0;
	if ((z / t) <= -1e-70)
		tmp = t_1;
	elseif ((z / t) <= 2e-17)
		tmp = x;
	elseif ((z / t) <= 5e+59)
		tmp = t_1;
	elseif (((z / t) <= 5e+135) || ~(((z / t) <= 5e+184)))
		tmp = (z / t) * -x;
	else
		tmp = z * (y / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -1e-70], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 2e-17], x, If[LessEqual[N[(z / t), $MachinePrecision], 5e+59], t$95$1, If[Or[LessEqual[N[(z / t), $MachinePrecision], 5e+135], N[Not[LessEqual[N[(z / t), $MachinePrecision], 5e+184]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 z t) < -9.99999999999999996e-71 or 2.00000000000000014e-17 < (/.f64 z t) < 4.9999999999999997e59

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. *-commutative 98.7%

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

      3. div-inv 97.4%

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

      4. associate-*l* 87.2%

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

      5. fma-def 87.3%

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

   3. Applied egg-rr 87.3%

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

   4. Taylor expanded in y around inf 59.6%

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

      1. associate-*r/ 66.2%

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

   6. Simplified 66.2%

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

   if -9.99999999999999996e-71 < (/.f64 z t) < 2.00000000000000014e-17

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 80.2%

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

   if 4.9999999999999997e59 < (/.f64 z t) < 5.00000000000000029e135 or 4.9999999999999999e184 < (/.f64 z t)

   1. Initial program 89.6%

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

   2. Taylor expanded in x around inf 67.3%

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

      1. *-commutative 67.3%

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

      2. mul-1-neg 67.3%

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

      3. unsub-neg 67.3%

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

      4. distribute-lft-out-- 67.3%

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

      5. *-rgt-identity 67.3%

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

   4. Simplified 67.3%

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

   5. Taylor expanded in x around 0 67.3%

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

   6. Taylor expanded in z around inf 63.7%

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

      1. associate-*l/ 67.3%

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

      2. associate-*r* 67.3%

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

      3. mul-1-neg 67.3%

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

      4. distribute-frac-neg 67.3%

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

      5. *-commutative 67.3%

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

   8. Simplified 67.3%

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

   if 5.00000000000000029e135 < (/.f64 z t) < 4.9999999999999999e184

   1. Initial program 99.5%

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

   2. Taylor expanded in z around -inf 85.2%

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

   3. Taylor expanded in t around 0 85.2%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+135} \lor \neg \left(\frac{z}{t} \leq 5 \cdot 10^{+184}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 3: 63.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{t} \cdot y\\ \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z t) y)))
   (if (<= (/ z t) -1e-70)
     t_1
     (if (<= (/ z t) 2e-17)
       x
       (if (<= (/ z t) 5e+59)
         t_1
         (if (<= (/ z t) 5e+135)
           (* (/ z t) (- x))
           (if (<= (/ z t) 5e+184) (* z (/ y t)) (* (- z) (/ x t)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = t_1;
	} else if ((z / t) <= 2e-17) {
		tmp = x;
	} else if ((z / t) <= 5e+59) {
		tmp = t_1;
	} else if ((z / t) <= 5e+135) {
		tmp = (z / t) * -x;
	} else if ((z / t) <= 5e+184) {
		tmp = z * (y / t);
	} else {
		tmp = -z * (x / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z / t) * y
    if ((z / t) <= (-1d-70)) then
        tmp = t_1
    else if ((z / t) <= 2d-17) then
        tmp = x
    else if ((z / t) <= 5d+59) then
        tmp = t_1
    else if ((z / t) <= 5d+135) then
        tmp = (z / t) * -x
    else if ((z / t) <= 5d+184) then
        tmp = z * (y / t)
    else
        tmp = -z * (x / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z / t) * y;
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = t_1;
	} else if ((z / t) <= 2e-17) {
		tmp = x;
	} else if ((z / t) <= 5e+59) {
		tmp = t_1;
	} else if ((z / t) <= 5e+135) {
		tmp = (z / t) * -x;
	} else if ((z / t) <= 5e+184) {
		tmp = z * (y / t);
	} else {
		tmp = -z * (x / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / t) * y
	tmp = 0
	if (z / t) <= -1e-70:
		tmp = t_1
	elif (z / t) <= 2e-17:
		tmp = x
	elif (z / t) <= 5e+59:
		tmp = t_1
	elif (z / t) <= 5e+135:
		tmp = (z / t) * -x
	elif (z / t) <= 5e+184:
		tmp = z * (y / t)
	else:
		tmp = -z * (x / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / t) * y)
	tmp = 0.0
	if (Float64(z / t) <= -1e-70)
		tmp = t_1;
	elseif (Float64(z / t) <= 2e-17)
		tmp = x;
	elseif (Float64(z / t) <= 5e+59)
		tmp = t_1;
	elseif (Float64(z / t) <= 5e+135)
		tmp = Float64(Float64(z / t) * Float64(-x));
	elseif (Float64(z / t) <= 5e+184)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = Float64(Float64(-z) * Float64(x / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / t) * y;
	tmp = 0.0;
	if ((z / t) <= -1e-70)
		tmp = t_1;
	elseif ((z / t) <= 2e-17)
		tmp = x;
	elseif ((z / t) <= 5e+59)
		tmp = t_1;
	elseif ((z / t) <= 5e+135)
		tmp = (z / t) * -x;
	elseif ((z / t) <= 5e+184)
		tmp = z * (y / t);
	else
		tmp = -z * (x / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[N[(z / t), $MachinePrecision], -1e-70], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 2e-17], x, If[LessEqual[N[(z / t), $MachinePrecision], 5e+59], t$95$1, If[LessEqual[N[(z / t), $MachinePrecision], 5e+135], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[N[(z / t), $MachinePrecision], 5e+184], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], N[((-z) * N[(x / t), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 z t) < -9.99999999999999996e-71 or 2.00000000000000014e-17 < (/.f64 z t) < 4.9999999999999997e59

   1. Initial program 98.7%

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

      1. +-commutative 98.7%

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

      2. *-commutative 98.7%

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

      3. div-inv 97.4%

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

      4. associate-*l* 87.2%

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

      5. fma-def 87.3%

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

   3. Applied egg-rr 87.3%

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

   4. Taylor expanded in y around inf 59.6%

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

      1. associate-*r/ 66.2%

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

   6. Simplified 66.2%

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

   if -9.99999999999999996e-71 < (/.f64 z t) < 2.00000000000000014e-17

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 80.2%

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

   if 4.9999999999999997e59 < (/.f64 z t) < 5.00000000000000029e135

   1. Initial program 99.8%

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

   2. Taylor expanded in x around inf 63.1%

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

      1. *-commutative 63.1%

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

      2. mul-1-neg 63.1%

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

      3. unsub-neg 63.1%

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

      4. distribute-lft-out-- 63.1%

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

      5. *-rgt-identity 63.1%

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

   4. Simplified 63.1%

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

   5. Taylor expanded in x around 0 63.1%

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

   6. Taylor expanded in z around inf 46.0%

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

      1. associate-*l/ 63.1%

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

      2. associate-*r* 63.1%

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

      3. mul-1-neg 63.1%

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

      4. distribute-frac-neg 63.1%

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

      5. *-commutative 63.1%

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

   8. Simplified 63.1%

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

   if 5.00000000000000029e135 < (/.f64 z t) < 4.9999999999999999e184

   1. Initial program 99.5%

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

   2. Taylor expanded in z around -inf 85.2%

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

   3. Taylor expanded in t around 0 85.2%

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

      1. associate-/l* 99.7%

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

      2. associate-/r/ 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in y around inf 100.0%

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

   if 4.9999999999999999e184 < (/.f64 z t)

   1. Initial program 85.1%

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

   2. Taylor expanded in x around inf 69.1%

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

      1. *-commutative 69.1%

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

      2. mul-1-neg 69.1%

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

      3. unsub-neg 69.1%

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

      4. distribute-lft-out-- 69.1%

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

      5. *-rgt-identity 69.1%

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

   4. Simplified 69.1%

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

   5. Taylor expanded in x around 0 69.1%

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

   6. Taylor expanded in z around inf 71.3%

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

      1. associate-*r/ 71.3%

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

      2. neg-mul-1 71.3%

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

      3. distribute-lft-neg-in 71.3%

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

      4. *-commutative 71.3%

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

   8. Simplified 71.3%

      \[\leadsto \color{blue}{\frac{x}{t} \cdot \left(-z\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 2 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+59}:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+135}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 5 \cdot 10^{+184}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\left(-z\right) \cdot \frac{x}{t}\\ \end{array} \]

Alternative 4: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70} \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -1e-70) (not (<= (/ z t) 2e-17))) (* (/ z t) (- y x)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -1e-70) || !((z / t) <= 2e-17)) {
		tmp = (z / t) * (y - x);
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z / t) <= (-1d-70)) .or. (.not. ((z / t) <= 2d-17))) then
        tmp = (z / t) * (y - x)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -1e-70) || !((z / t) <= 2e-17)) {
		tmp = (z / t) * (y - x);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -1e-70) or not ((z / t) <= 2e-17):
		tmp = (z / t) * (y - x)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -1e-70) || !(Float64(z / t) <= 2e-17))
		tmp = Float64(Float64(z / t) * Float64(y - x));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -1e-70) || ~(((z / t) <= 2e-17)))
		tmp = (z / t) * (y - x);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -1e-70], N[Not[LessEqual[N[(z / t), $MachinePrecision], 2e-17]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -9.99999999999999996e-71 or 2.00000000000000014e-17 < (/.f64 z t)

   1. Initial program 95.4%

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

   2. Taylor expanded in z around -inf 91.4%

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

   3. Taylor expanded in t around 0 87.9%

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

      1. div-inv 87.8%

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

      2. *-commutative 87.8%

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

      3. associate-*r* 89.4%

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

      4. *-commutative 89.4%

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

      5. associate-*r* 91.7%

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

      6. div-inv 91.8%

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

   5. Applied egg-rr 91.8%

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

   if -9.99999999999999996e-71 < (/.f64 z t) < 2.00000000000000014e-17

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 80.2%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70} \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5: 84.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 1000000:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -1e-70)
   (* (/ z t) (- y x))
   (if (<= (/ z t) 1000000.0) (* x (- 1.0 (/ z t))) (* z (/ (- y x) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = (z / t) * (y - x);
	} else if ((z / t) <= 1000000.0) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z / t) <= (-1d-70)) then
        tmp = (z / t) * (y - x)
    else if ((z / t) <= 1000000.0d0) then
        tmp = x * (1.0d0 - (z / t))
    else
        tmp = z * ((y - x) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -1e-70) {
		tmp = (z / t) * (y - x);
	} else if ((z / t) <= 1000000.0) {
		tmp = x * (1.0 - (z / t));
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= -1e-70:
		tmp = (z / t) * (y - x)
	elif (z / t) <= 1000000.0:
		tmp = x * (1.0 - (z / t))
	else:
		tmp = z * ((y - x) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= -1e-70)
		tmp = Float64(Float64(z / t) * Float64(y - x));
	elseif (Float64(z / t) <= 1000000.0)
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	else
		tmp = Float64(z * Float64(Float64(y - x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= -1e-70)
		tmp = (z / t) * (y - x);
	elseif ((z / t) <= 1000000.0)
		tmp = x * (1.0 - (z / t));
	else
		tmp = z * ((y - x) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], -1e-70], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z / t), $MachinePrecision], 1000000.0], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 z t) < -9.99999999999999996e-71

   1. Initial program 98.4%

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

   2. Taylor expanded in z around -inf 91.3%

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

   3. Taylor expanded in t around 0 85.7%

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

      1. div-inv 85.5%

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

      2. *-commutative 85.5%

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

      3. associate-*r* 85.4%

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

      4. *-commutative 85.4%

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

      5. associate-*r* 92.7%

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

      6. div-inv 92.8%

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

   5. Applied egg-rr 92.8%

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

   if -9.99999999999999996e-71 < (/.f64 z t) < 1e6

   1. Initial program 99.1%

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

   2. Taylor expanded in x around inf 79.3%

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

      1. *-commutative 79.3%

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

      2. mul-1-neg 79.3%

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

      3. unsub-neg 79.3%

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

      4. distribute-lft-out-- 79.3%

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

      5. *-rgt-identity 79.3%

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

   4. Simplified 79.3%

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

   5. Taylor expanded in x around 0 79.3%

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

   if 1e6 < (/.f64 z t)

   1. Initial program 92.5%

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

   2. Taylor expanded in z around -inf 91.2%

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

   3. Taylor expanded in t around 0 91.2%

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

      1. associate-/l* 92.5%

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

      2. associate-/r/ 95.7%

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

   5. Applied egg-rr 95.7%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 1000000:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 6: 95.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 1:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) -4e+24)
   (* (/ z t) (- y x))
   (if (<= (/ z t) 1.0) (+ x (* (/ z t) y)) (* z (/ (- y x) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -4e+24) {
		tmp = (z / t) * (y - x);
	} else if ((z / t) <= 1.0) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z / t) <= (-4d+24)) then
        tmp = (z / t) * (y - x)
    else if ((z / t) <= 1.0d0) then
        tmp = x + ((z / t) * y)
    else
        tmp = z * ((y - x) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= -4e+24) {
		tmp = (z / t) * (y - x);
	} else if ((z / t) <= 1.0) {
		tmp = x + ((z / t) * y);
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= -4e+24:
		tmp = (z / t) * (y - x)
	elif (z / t) <= 1.0:
		tmp = x + ((z / t) * y)
	else:
		tmp = z * ((y - x) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= -4e+24)
		tmp = Float64(Float64(z / t) * Float64(y - x));
	elseif (Float64(z / t) <= 1.0)
		tmp = Float64(x + Float64(Float64(z / t) * y));
	else
		tmp = Float64(z * Float64(Float64(y - x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= -4e+24)
		tmp = (z / t) * (y - x);
	elseif ((z / t) <= 1.0)
		tmp = x + ((z / t) * y);
	else
		tmp = z * ((y - x) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], -4e+24], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z / t), $MachinePrecision], 1.0], N[(x + N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 z t) < -3.9999999999999999e24

   1. Initial program 98.0%

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

   2. Taylor expanded in z around -inf 94.4%

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

   3. Taylor expanded in t around 0 94.4%

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

      1. div-inv 94.3%

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

      2. *-commutative 94.3%

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

      3. associate-*r* 94.1%

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

      4. *-commutative 94.1%

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

      5. associate-*r* 97.9%

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

      6. div-inv 98.0%

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

   5. Applied egg-rr 98.0%

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

   if -3.9999999999999999e24 < (/.f64 z t) < 1

   1. Initial program 99.2%

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

   2. Taylor expanded in y around inf 93.9%

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

      1. associate-*r/ 98.4%

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

   4. Simplified 98.4%

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

   if 1 < (/.f64 z t)

   1. Initial program 92.6%

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

   2. Taylor expanded in z around -inf 91.3%

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

   3. Taylor expanded in t around 0 90.6%

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

      1. associate-/l* 91.9%

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

      2. associate-/r/ 95.1%

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

   5. Applied egg-rr 95.1%

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

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -4 \cdot 10^{+24}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{elif}\;\frac{z}{t} \leq 1:\\ \;\;\;\;x + \frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 7: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70} \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (/ z t) -1e-70) (not (<= (/ z t) 2e-17))) (* (/ z t) y) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -1e-70) || !((z / t) <= 2e-17)) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z / t) <= (-1d-70)) .or. (.not. ((z / t) <= 2d-17))) then
        tmp = (z / t) * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z / t) <= -1e-70) || !((z / t) <= 2e-17)) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z / t) <= -1e-70) or not ((z / t) <= 2e-17):
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z / t) <= -1e-70) || !(Float64(z / t) <= 2e-17))
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z / t) <= -1e-70) || ~(((z / t) <= 2e-17)))
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z / t), $MachinePrecision], -1e-70], N[Not[LessEqual[N[(z / t), $MachinePrecision], 2e-17]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < -9.99999999999999996e-71 or 2.00000000000000014e-17 < (/.f64 z t)

   1. Initial program 95.4%

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

      1. +-commutative 95.4%

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

      2. *-commutative 95.4%

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

      3. div-inv 94.5%

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

      4. associate-*l* 91.7%

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

      5. fma-def 91.7%

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

   3. Applied egg-rr 91.7%

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

   4. Taylor expanded in y around inf 55.7%

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

      1. associate-*r/ 60.3%

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

   6. Simplified 60.3%

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

   if -9.99999999999999996e-71 < (/.f64 z t) < 2.00000000000000014e-17

   1. Initial program 99.1%

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

   2. Taylor expanded in z around 0 80.2%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq -1 \cdot 10^{-70} \lor \neg \left(\frac{z}{t} \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ z t) 5e+134) (+ x (* (/ z t) (- y x))) (* z (/ (- y x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 5e+134) {
		tmp = x + ((z / t) * (y - x));
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z / t) <= 5d+134) then
        tmp = x + ((z / t) * (y - x))
    else
        tmp = z * ((y - x) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z / t) <= 5e+134) {
		tmp = x + ((z / t) * (y - x));
	} else {
		tmp = z * ((y - x) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z / t) <= 5e+134:
		tmp = x + ((z / t) * (y - x))
	else:
		tmp = z * ((y - x) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z / t) <= 5e+134)
		tmp = Float64(x + Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = Float64(z * Float64(Float64(y - x) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z / t) <= 5e+134)
		tmp = x + ((z / t) * (y - x));
	else
		tmp = z * ((y - x) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(z / t), $MachinePrecision], 5e+134], N[(x + N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z t) < 4.99999999999999981e134

   1. Initial program 99.0%

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

   if 4.99999999999999981e134 < (/.f64 z t)

   1. Initial program 87.4%

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

   2. Taylor expanded in z around -inf 97.9%

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

   3. Taylor expanded in t around 0 97.9%

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

      1. associate-/l* 87.4%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z}{t} \leq 5 \cdot 10^{+134}:\\ \;\;\;\;x + \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \end{array} \]

Alternative 9: 71.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-16} \lor \neg \left(y \leq 1.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.6e-16) (not (<= y 1.9e-19)))
   (* (/ z t) y)
   (* x (- 1.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.6e-16) || !(y <= 1.9e-19)) {
		tmp = (z / t) * y;
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-2.6d-16)) .or. (.not. (y <= 1.9d-19))) then
        tmp = (z / t) * y
    else
        tmp = x * (1.0d0 - (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.6e-16) || !(y <= 1.9e-19)) {
		tmp = (z / t) * y;
	} else {
		tmp = x * (1.0 - (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.6e-16) or not (y <= 1.9e-19):
		tmp = (z / t) * y
	else:
		tmp = x * (1.0 - (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.6e-16) || !(y <= 1.9e-19))
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = Float64(x * Float64(1.0 - Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -2.6e-16) || ~((y <= 1.9e-19)))
		tmp = (z / t) * y;
	else
		tmp = x * (1.0 - (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.6e-16], N[Not[LessEqual[y, 1.9e-19]], $MachinePrecision]], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], N[(x * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.5999999999999998e-16 or 1.9e-19 < y

   1. Initial program 99.1%

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

      1. +-commutative 99.1%

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

      2. *-commutative 99.1%

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

      3. div-inv 99.0%

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

      4. associate-*l* 91.4%

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

      5. fma-def 91.4%

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

   3. Applied egg-rr 91.4%

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

   4. Taylor expanded in y around inf 61.9%

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

      1. associate-*r/ 68.5%

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

   6. Simplified 68.5%

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

   if -2.5999999999999998e-16 < y < 1.9e-19

   1. Initial program 94.9%

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

   2. Taylor expanded in x around inf 85.3%

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

      1. *-commutative 85.3%

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

      2. mul-1-neg 85.3%

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

      3. unsub-neg 85.3%

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

      4. distribute-lft-out-- 85.3%

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

      5. *-rgt-identity 85.3%

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

   4. Simplified 85.3%

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

   5. Taylor expanded in x around 0 85.3%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-16} \lor \neg \left(y \leq 1.9 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{t}\right)\\ \end{array} \]

Alternative 10: 37.5% accurate, 9.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 97.0%

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

2. Taylor expanded in z around 0 38.0%

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

3. Final simplification 38.0%

   \[\leadsto x \]

Developer target: 97.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) \cdot \frac{z}{t}\\ t_2 := x + \frac{y - x}{\frac{t}{z}}\\ \mathbf{if}\;t_1 < -1013646692435.8867:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 < 0:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y x) (/ z t))) (t_2 (+ x (/ (- y x) (/ t z)))))
   (if (< t_1 -1013646692435.8867)
     t_2
     (if (< t_1 0.0) (+ x (/ (* (- y x) z) t)) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = x + ((y - x) / (t / z));
	double tmp;
	if (t_1 < -1013646692435.8867) {
		tmp = t_2;
	} else if (t_1 < 0.0) {
		tmp = x + (((y - x) * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y - x) * (z / t)
    t_2 = x + ((y - x) / (t / z))
    if (t_1 < (-1013646692435.8867d0)) then
        tmp = t_2
    else if (t_1 < 0.0d0) then
        tmp = x + (((y - x) * z) / t)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - x) * (z / t);
	double t_2 = x + ((y - x) / (t / z));
	double tmp;
	if (t_1 < -1013646692435.8867) {
		tmp = t_2;
	} else if (t_1 < 0.0) {
		tmp = x + (((y - x) * z) / t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - x) * (z / t)
	t_2 = x + ((y - x) / (t / z))
	tmp = 0
	if t_1 < -1013646692435.8867:
		tmp = t_2
	elif t_1 < 0.0:
		tmp = x + (((y - x) * z) / t)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - x) * Float64(z / t))
	t_2 = Float64(x + Float64(Float64(y - x) / Float64(t / z)))
	tmp = 0.0
	if (t_1 < -1013646692435.8867)
		tmp = t_2;
	elseif (t_1 < 0.0)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / t));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - x) * (z / t);
	t_2 = x + ((y - x) / (t / z));
	tmp = 0.0;
	if (t_1 < -1013646692435.8867)
		tmp = t_2;
	elseif (t_1 < 0.0)
		tmp = x + (((y - x) * z) / t);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] * N[(z / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$1, -1013646692435.8867], t$95$2, If[Less[t$95$1, 0.0], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z t)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:tickPosition from plot-0.2.3.4"
  :precision binary64

  :herbie-target
  (if (< (* (- y x) (/ z t)) -1013646692435.8867) (+ x (/ (- y x) (/ t z))) (if (< (* (- y x) (/ z t)) 0.0) (+ x (/ (* (- y x) z) t)) (+ x (/ (- y x) (/ t z)))))

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